| L(s) = 1 | − 24·7-s − 24·9-s + 192·23-s + 4·25-s + 96·41-s − 384·47-s + 56·49-s + 576·63-s + 324·81-s + 384·89-s + 696·103-s − 1.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.60e3·161-s + 163-s + 167-s − 856·169-s + 173-s − 96·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.42·7-s − 8/3·9-s + 8.34·23-s + 4/25·25-s + 2.34·41-s − 8.17·47-s + 8/7·49-s + 64/7·63-s + 4·81-s + 4.31·89-s + 6.75·103-s − 11.5·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 28.6·161-s + 0.00613·163-s + 0.00598·167-s − 5.06·169-s + 0.00578·173-s − 0.548·175-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.005652179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005652179\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{8} \) |
| 5 | \( 1 - 4 T^{2} - 1388 T^{4} + 2948 T^{6} + 1654 p^{4} T^{8} + 2948 p^{4} T^{10} - 1388 p^{8} T^{12} - 4 p^{12} T^{14} + p^{16} T^{16} \) |
| good | 7 | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + 1446 T^{4} + 6 p^{3} T^{5} + 76 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 11 | \( ( 1 + 700 T^{2} + 236116 T^{4} + 49768132 T^{6} + 7187557270 T^{8} + 49768132 p^{4} T^{10} + 236116 p^{8} T^{12} + 700 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 + 428 T^{2} + 93364 T^{4} + 21059924 T^{6} + 4286842966 T^{8} + 21059924 p^{4} T^{10} + 93364 p^{8} T^{12} + 428 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 - 1484 T^{2} + 1080004 T^{4} - 516635636 T^{6} + 175901845750 T^{8} - 516635636 p^{4} T^{10} + 1080004 p^{8} T^{12} - 1484 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 + 1232 T^{2} + 578332 T^{4} + 132768560 T^{6} + 27078201862 T^{8} + 132768560 p^{4} T^{10} + 578332 p^{8} T^{12} + 1232 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 - 48 T + 2632 T^{2} - 72144 T^{3} + 2159694 T^{4} - 72144 p^{2} T^{5} + 2632 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 29 | \( ( 1 - 532 T^{2} + 311284 T^{4} - 421621996 T^{6} + 1070403305686 T^{8} - 421621996 p^{4} T^{10} + 311284 p^{8} T^{12} - 532 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 - 640 T^{2} + 1597948 T^{4} - 1080568192 T^{6} + 2370900390406 T^{8} - 1080568192 p^{4} T^{10} + 1597948 p^{8} T^{12} - 640 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 + 1916 T^{2} + 7794964 T^{4} + 10130228228 T^{6} + 21975260876566 T^{8} + 10130228228 p^{4} T^{10} + 7794964 p^{8} T^{12} + 1916 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 24 T + 3052 T^{2} - 127080 T^{3} + 6081126 T^{4} - 127080 p^{2} T^{5} + 3052 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 - 5528 T^{2} + 22565692 T^{4} - 58914945320 T^{6} + 128128709162182 T^{8} - 58914945320 p^{4} T^{10} + 22565692 p^{8} T^{12} - 5528 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 96 T + 8728 T^{2} + 420192 T^{3} + 24060270 T^{4} + 420192 p^{2} T^{5} + 8728 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 53 | \( ( 1 + 9260 T^{2} + 50054836 T^{4} + 180838486868 T^{6} + 196845209350 p^{2} T^{8} + 180838486868 p^{4} T^{10} + 50054836 p^{8} T^{12} + 9260 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 + 10828 T^{2} + 79922740 T^{4} + 407302348084 T^{6} + 1632709190053078 T^{8} + 407302348084 p^{4} T^{10} + 79922740 p^{8} T^{12} + 10828 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 14888 T^{2} + 116042524 T^{4} - 638729972120 T^{6} + 2706013142650246 T^{8} - 638729972120 p^{4} T^{10} + 116042524 p^{8} T^{12} - 14888 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 - 392 p T^{2} + 331246012 T^{4} - 2603080621352 T^{6} + 13970293735045702 T^{8} - 2603080621352 p^{4} T^{10} + 331246012 p^{8} T^{12} - 392 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 17384 T^{2} + 155712988 T^{4} - 1120047248216 T^{6} + 1299632274982 p^{2} T^{8} - 1120047248216 p^{4} T^{10} + 155712988 p^{8} T^{12} - 17384 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 35432 T^{2} + 581836252 T^{4} - 5751550649048 T^{6} + 37370671240483654 T^{8} - 5751550649048 p^{4} T^{10} + 581836252 p^{8} T^{12} - 35432 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 42688 T^{2} + 823372156 T^{4} - 9479114483008 T^{6} + 71897126801102854 T^{8} - 9479114483008 p^{4} T^{10} + 823372156 p^{8} T^{12} - 42688 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 83 | \( ( 1 - 16472 T^{2} + 235980604 T^{4} - 2293408901480 T^{6} + 17688767095712326 T^{8} - 2293408901480 p^{4} T^{10} + 235980604 p^{8} T^{12} - 16472 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 96 T + 24316 T^{2} - 1287072 T^{3} + 235064262 T^{4} - 1287072 p^{2} T^{5} + 24316 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 344 p T^{2} + 531297532 T^{4} - 6322502759144 T^{6} + 64825056663850630 T^{8} - 6322502759144 p^{4} T^{10} + 531297532 p^{8} T^{12} - 344 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.43557943378103011098705327967, −2.40310207418277672447324894120, −2.35006426793582200827293392021, −2.22711350332699723081286068260, −2.21785278570936940866995184745, −1.89239003196016311464589995976, −1.86735038703120265449164163468, −1.86476477545499172698669086440, −1.53402561176277492960944510448, −1.50140351557859593720352275687, −1.42684598078448248351044334547, −1.29728145018785650111713431923, −1.28926773879745971037437433720, −1.23570420660288951946061189139, −1.22607135735235356982795858535, −1.18317406398135897428752883901, −1.10304851219722337432784903086, −0.75264395607057947793490944450, −0.53220892994172704986099601338, −0.52466866332136239356895428706, −0.51762854056801749886251283170, −0.43500303305723528298846234392, −0.23954425358455670060248526210, −0.17305729539976940259960840442, −0.12935598760945331997732704073,
0.12935598760945331997732704073, 0.17305729539976940259960840442, 0.23954425358455670060248526210, 0.43500303305723528298846234392, 0.51762854056801749886251283170, 0.52466866332136239356895428706, 0.53220892994172704986099601338, 0.75264395607057947793490944450, 1.10304851219722337432784903086, 1.18317406398135897428752883901, 1.22607135735235356982795858535, 1.23570420660288951946061189139, 1.28926773879745971037437433720, 1.29728145018785650111713431923, 1.42684598078448248351044334547, 1.50140351557859593720352275687, 1.53402561176277492960944510448, 1.86476477545499172698669086440, 1.86735038703120265449164163468, 1.89239003196016311464589995976, 2.21785278570936940866995184745, 2.22711350332699723081286068260, 2.35006426793582200827293392021, 2.40310207418277672447324894120, 2.43557943378103011098705327967
Plot not available for L-functions of degree greater than 10.
