| L(s) = 1 | + 2·2-s − 8·3-s − 5·5-s − 16·6-s − 8·8-s + 27·9-s − 10·10-s − 68·11-s − 68·13-s + 40·15-s − 16·16-s + 74·17-s + 54·18-s − 128·19-s − 136·22-s + 80·23-s + 64·24-s − 136·26-s − 136·27-s + 572·29-s + 80·30-s − 24·31-s + 544·33-s + 148·34-s − 294·37-s − 256·38-s + 544·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s − 0.447·5-s − 1.08·6-s − 0.353·8-s + 9-s − 0.316·10-s − 1.86·11-s − 1.45·13-s + 0.688·15-s − 1/4·16-s + 1.05·17-s + 0.707·18-s − 1.54·19-s − 1.31·22-s + 0.725·23-s + 0.544·24-s − 1.02·26-s − 0.969·27-s + 3.66·29-s + 0.486·30-s − 0.139·31-s + 2.86·33-s + 0.746·34-s − 1.30·37-s − 1.09·38-s + 2.23·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1352030391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1352030391\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 8 T + 37 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 68 T + 3293 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 74 T + 563 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 128 T + 9525 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 80 T - 5767 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 286 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 24 T - 29215 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 294 T + 35783 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 312 T - 6479 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 34 T - 147721 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 168 T - 177155 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 170 T - 198081 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 564 T + 17333 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 616 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 250 T - 326517 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 944 T + 398097 T^{2} - 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 672 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1430 T + 1339931 T^{2} + 1430 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1270 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61700530524657880433539397650, −10.35029174515526122436598760404, −10.32139278051352240591238492224, −9.733084948197398240755343733818, −8.889942273013904019319057060852, −8.438627845440572139599807026080, −7.943052278604753108627977826230, −7.58646586992676928057198763710, −6.81587846366926063633504034523, −6.62431362090390501237188221767, −6.02179859419833520947823418522, −5.31955511391296798737809022707, −5.07995387512281360000993068421, −4.90603317107726189151037830979, −4.25435309721528595591569604460, −3.48485678526724594609695455488, −2.71209040174578898634372992446, −2.35302027481907729330631931392, −1.01558961997591997477331378189, −0.13372096065715616314592159822,
0.13372096065715616314592159822, 1.01558961997591997477331378189, 2.35302027481907729330631931392, 2.71209040174578898634372992446, 3.48485678526724594609695455488, 4.25435309721528595591569604460, 4.90603317107726189151037830979, 5.07995387512281360000993068421, 5.31955511391296798737809022707, 6.02179859419833520947823418522, 6.62431362090390501237188221767, 6.81587846366926063633504034523, 7.58646586992676928057198763710, 7.943052278604753108627977826230, 8.438627845440572139599807026080, 8.889942273013904019319057060852, 9.733084948197398240755343733818, 10.32139278051352240591238492224, 10.35029174515526122436598760404, 10.61700530524657880433539397650
