| L(s) = 1 | − 6·9-s − 16·13-s + 48·17-s + 48·29-s + 160·37-s + 168·41-s + 52·49-s + 192·53-s + 344·61-s − 88·73-s + 27·81-s − 264·89-s + 344·97-s + 720·101-s − 296·109-s + 96·113-s + 96·117-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 288·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 1.23·13-s + 2.82·17-s + 1.65·29-s + 4.32·37-s + 4.09·41-s + 1.06·49-s + 3.62·53-s + 5.63·61-s − 1.20·73-s + 1/3·81-s − 2.96·89-s + 3.54·97-s + 7.12·101-s − 2.71·109-s + 0.849·113-s + 0.820·117-s + 2.80·121-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 − 1.88·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.85491734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.85491734\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2598 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 24 T + 542 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 580 T^{2} + 160422 T^{4} - 580 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1540 T^{2} + 1106502 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 206 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2980 T^{2} + 3882822 T^{4} - 2980 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 80 T + 4158 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 + 380 T^{2} + 6136422 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6532 T^{2} + 19688838 T^{4} - 6532 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 96 T + 6302 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10900 T^{2} + 53588742 T^{4} - 10900 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 172 T + 14118 T^{2} - 172 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12100 T^{2} + 70269222 T^{4} - 12100 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 3452 T^{2} + 51544518 T^{4} + 3452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 44 T + 4662 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12242 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9412 T^{2} + 57343398 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 132 T + 19478 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 172 T + 23334 T^{2} - 172 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.92129365393768373818402297196, −6.42812475470735212054041642662, −6.07885587498022083438704556789, −6.04006325126179651838228796486, −6.03774128105043014418913924145, −5.51032423594509161105775176854, −5.47481346048837658251010457853, −5.45318064033465931435727987333, −5.02645162899157424396197638739, −4.50676030945575431262642645611, −4.44339206843992525112561610912, −4.40997371042790358524885752465, −4.08392831726580787445734147682, −3.63151412852490608140464249459, −3.41083337975440712382313694888, −3.36467111030648667091806975290, −2.71142721468103668879598543355, −2.51749279890710703234685855653, −2.46317239087533692177992581327, −2.38447155884635029685521417135, −1.89227992603190522825187849984, −0.897310002391564029658440084319, −0.877108810427134899606980037207, −0.847888882755244096955368521552, −0.69077345616103996322672489175,
0.69077345616103996322672489175, 0.847888882755244096955368521552, 0.877108810427134899606980037207, 0.897310002391564029658440084319, 1.89227992603190522825187849984, 2.38447155884635029685521417135, 2.46317239087533692177992581327, 2.51749279890710703234685855653, 2.71142721468103668879598543355, 3.36467111030648667091806975290, 3.41083337975440712382313694888, 3.63151412852490608140464249459, 4.08392831726580787445734147682, 4.40997371042790358524885752465, 4.44339206843992525112561610912, 4.50676030945575431262642645611, 5.02645162899157424396197638739, 5.45318064033465931435727987333, 5.47481346048837658251010457853, 5.51032423594509161105775176854, 6.03774128105043014418913924145, 6.04006325126179651838228796486, 6.07885587498022083438704556789, 6.42812475470735212054041642662, 6.92129365393768373818402297196
