| L(s) = 1 | + 4·5-s + 38·9-s − 12·11-s + 140·19-s − 109·25-s + 500·29-s + 680·31-s + 116·41-s + 152·45-s + 670·49-s − 48·55-s + 684·59-s − 1.11e3·61-s − 2.01e3·71-s + 1.64e3·79-s + 715·81-s − 3.10e3·89-s + 560·95-s − 456·99-s + 1.94e3·101-s − 824·109-s − 2.55e3·121-s − 936·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 1.40·9-s − 0.328·11-s + 1.69·19-s − 0.871·25-s + 3.20·29-s + 3.93·31-s + 0.441·41-s + 0.503·45-s + 1.95·49-s − 0.117·55-s + 1.50·59-s − 2.34·61-s − 3.36·71-s + 2.34·79-s + 0.980·81-s − 3.69·89-s + 0.604·95-s − 0.462·99-s + 1.91·101-s − 0.724·109-s − 1.91·121-s − 0.669·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.827864443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.827864443\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3630 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 13930 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 340 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 58 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 11558 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 164382 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163798 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 558 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 226982 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1008 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 153934 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 824 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 84090 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1550 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 715490 T^{2} + p^{6} T^{4} \) |
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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
−11.69145510926906354899823847883, −11.66036044295172971438133269448, −10.43831598433092614628779229120, −10.31289184307295641083777053182, −10.02375114020613949890291436866, −9.613334467352569704345481868748, −8.855628586616127282940810591898, −8.412913970409179472145473124083, −7.73792789908339474099336073325, −7.49839119447699598374479338734, −6.69143923594913703680838220206, −6.39746177745593526683111432266, −5.76077550663047662022118430051, −4.94297772926809496681537778391, −4.54091259396278321628343865549, −4.05459994161646137843373693602, −2.83442221009130971829247767048, −2.69275407997802958341846710940, −1.26920872295957378045838603256, −0.941126694262696290346152887810,
0.941126694262696290346152887810, 1.26920872295957378045838603256, 2.69275407997802958341846710940, 2.83442221009130971829247767048, 4.05459994161646137843373693602, 4.54091259396278321628343865549, 4.94297772926809496681537778391, 5.76077550663047662022118430051, 6.39746177745593526683111432266, 6.69143923594913703680838220206, 7.49839119447699598374479338734, 7.73792789908339474099336073325, 8.412913970409179472145473124083, 8.855628586616127282940810591898, 9.613334467352569704345481868748, 10.02375114020613949890291436866, 10.31289184307295641083777053182, 10.43831598433092614628779229120, 11.66036044295172971438133269448, 11.69145510926906354899823847883
