| L(s) = 1 | + 8.48·3-s + 9.89·5-s − 171·9-s − 308·11-s + 912.·13-s + 83.9·15-s + 451.·17-s + 2.61e3·19-s + 2.32e3·23-s − 3.02e3·25-s − 3.51e3·27-s − 6.48e3·29-s − 814.·31-s − 2.61e3·33-s − 1.22e4·37-s + 7.73e3·39-s + 527.·41-s + 1.50e4·43-s − 1.69e3·45-s − 1.60e3·47-s + 3.82e3·51-s + 1.07e4·53-s − 3.04e3·55-s + 2.22e4·57-s + 5.21e4·59-s + 3.67e4·61-s + 9.03e3·65-s + ⋯ |
| L(s) = 1 | + 0.544·3-s + 0.177·5-s − 0.703·9-s − 0.767·11-s + 1.49·13-s + 0.0963·15-s + 0.378·17-s + 1.66·19-s + 0.916·23-s − 0.968·25-s − 0.927·27-s − 1.43·29-s − 0.152·31-s − 0.417·33-s − 1.46·37-s + 0.814·39-s + 0.0490·41-s + 1.23·43-s − 0.124·45-s − 0.105·47-s + 0.206·51-s + 0.527·53-s − 0.135·55-s + 0.905·57-s + 1.94·59-s + 1.26·61-s + 0.265·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.725242590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725242590\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8.48T + 243T^{2} \) |
| 5 | \( 1 - 9.89T + 3.12e3T^{2} \) |
| 11 | \( 1 + 308T + 1.61e5T^{2} \) |
| 13 | \( 1 - 912.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 451.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 814.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 527.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.68e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.376109690089221738620196383313, −8.730890795787387503429535839025, −7.896492188446096059429345023818, −7.13526606623301754465085000344, −5.74031137890457294700181505295, −5.39967001856919488712544965605, −3.76130432427487696044747427475, −3.13593826553810647364334658734, −1.96799729286528482948672336792, −0.73842641798343365085619624178,
0.73842641798343365085619624178, 1.96799729286528482948672336792, 3.13593826553810647364334658734, 3.76130432427487696044747427475, 5.39967001856919488712544965605, 5.74031137890457294700181505295, 7.13526606623301754465085000344, 7.896492188446096059429345023818, 8.730890795787387503429535839025, 9.376109690089221738620196383313
