| L(s) = 1 | + 2·2-s − 6·5-s − 8·8-s − 12·10-s + 30·11-s − 4·13-s − 16·16-s − 66·17-s − 52·19-s + 60·22-s + 114·23-s + 125·25-s − 8·26-s − 144·29-s − 196·31-s − 132·34-s + 286·37-s − 104·38-s + 48·40-s − 756·41-s + 328·43-s + 228·46-s + 228·47-s + 250·50-s − 348·53-s − 180·55-s − 288·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.536·5-s − 0.353·8-s − 0.379·10-s + 0.822·11-s − 0.0853·13-s − 1/4·16-s − 0.941·17-s − 0.627·19-s + 0.581·22-s + 1.03·23-s + 25-s − 0.0603·26-s − 0.922·29-s − 1.13·31-s − 0.665·34-s + 1.27·37-s − 0.443·38-s + 0.189·40-s − 2.87·41-s + 1.16·43-s + 0.730·46-s + 0.707·47-s + 0.707·50-s − 0.901·53-s − 0.441·55-s − 0.652·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8321770494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8321770494\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 66 T - 557 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 52 T - 4155 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 114 T + 829 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 196 T + 8625 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 286 T + 31143 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 164 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 228 T - 51839 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 348 T - 27773 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 348 T - 84275 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 106 T - 215745 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1042 T + 696747 T^{2} + 1042 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 88 T - 485295 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1440 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1374 T + 1182907 T^{2} + 1374 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
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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
−10.06946770712219123274139885853, −9.467618317287535754497551797318, −8.861101604941372303105554871631, −8.755853102314902768029984070298, −8.603458514274713797807768888849, −7.52703542281427559816487036403, −7.49537530553783123397942145595, −6.83186050129593487593997467383, −6.64804965104437394035672070558, −5.82817501837948010021124025733, −5.75976421998464897104921512349, −4.85242060712740558978671319614, −4.68981066610943421652523364459, −4.00390579259311710752307069156, −3.87562107281846589953842992237, −2.96320253420233940635106291917, −2.79364914463039710885903975291, −1.76244568606205722325927727087, −1.30001616275690227137719217520, −0.21270023740601513564615700859,
0.21270023740601513564615700859, 1.30001616275690227137719217520, 1.76244568606205722325927727087, 2.79364914463039710885903975291, 2.96320253420233940635106291917, 3.87562107281846589953842992237, 4.00390579259311710752307069156, 4.68981066610943421652523364459, 4.85242060712740558978671319614, 5.75976421998464897104921512349, 5.82817501837948010021124025733, 6.64804965104437394035672070558, 6.83186050129593487593997467383, 7.49537530553783123397942145595, 7.52703542281427559816487036403, 8.603458514274713797807768888849, 8.755853102314902768029984070298, 8.861101604941372303105554871631, 9.467618317287535754497551797318, 10.06946770712219123274139885853
