| L(s) = 1 | − 28.7·3-s + 25·5-s − 42.1·7-s + 581.·9-s + 416.·11-s + 966.·13-s − 717.·15-s − 1.83e3·17-s + 317.·19-s + 1.21e3·21-s + 1.56e3·23-s + 625·25-s − 9.72e3·27-s + 7.75e3·29-s + 102.·31-s − 1.19e4·33-s − 1.05e3·35-s + 1.93e3·37-s − 2.77e4·39-s + 7.99e3·41-s + 1.65e4·43-s + 1.45e4·45-s + 1.86e4·47-s − 1.50e4·49-s + 5.26e4·51-s − 1.49e4·53-s + 1.04e4·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 0.447·5-s − 0.325·7-s + 2.39·9-s + 1.03·11-s + 1.58·13-s − 0.823·15-s − 1.53·17-s + 0.201·19-s + 0.598·21-s + 0.618·23-s + 0.200·25-s − 2.56·27-s + 1.71·29-s + 0.0191·31-s − 1.91·33-s − 0.145·35-s + 0.232·37-s − 2.92·39-s + 0.742·41-s + 1.36·43-s + 1.07·45-s + 1.23·47-s − 0.894·49-s + 2.83·51-s − 0.732·53-s + 0.463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9871031467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9871031467\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 + 28.7T + 243T^{2} \) |
| 7 | \( 1 + 42.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 966.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 317.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 102.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.01e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e4T + 8.58e9T^{2} \) |
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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
−15.62375432497308448995676140324, −13.67907441729859442256179111506, −12.57677095638850143305280077938, −11.38187244956132201060069553240, −10.63299398244359297175836293002, −9.119816626608655154943870402951, −6.69028411269434210771731885993, −6.02982442502564268997390633906, −4.38756066350892397807912794873, −1.03926941274387637888507780149,
1.03926941274387637888507780149, 4.38756066350892397807912794873, 6.02982442502564268997390633906, 6.69028411269434210771731885993, 9.119816626608655154943870402951, 10.63299398244359297175836293002, 11.38187244956132201060069553240, 12.57677095638850143305280077938, 13.67907441729859442256179111506, 15.62375432497308448995676140324
