| L(s) = 1 | − 3.27e4·2-s + 1.07e7·3-s + 1.07e9·4-s − 3.05e10·5-s − 3.52e11·6-s + 2.26e13·7-s − 3.51e13·8-s − 5.02e14·9-s + 1.00e15·10-s + 2.45e16·11-s + 1.15e16·12-s − 1.48e17·13-s − 7.40e17·14-s − 3.28e17·15-s + 1.15e18·16-s + 1.16e19·17-s + 1.64e19·18-s + 3.78e19·19-s − 3.27e19·20-s + 2.42e20·21-s − 8.04e20·22-s + 8.75e20·23-s − 3.78e20·24-s + 9.31e20·25-s + 4.86e21·26-s − 1.20e22·27-s + 2.42e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.432·3-s + 0.5·4-s − 0.447·5-s − 0.305·6-s + 1.79·7-s − 0.353·8-s − 0.812·9-s + 0.316·10-s + 1.77·11-s + 0.216·12-s − 0.804·13-s − 1.27·14-s − 0.193·15-s + 0.250·16-s + 0.984·17-s + 0.574·18-s + 0.571·19-s − 0.223·20-s + 0.778·21-s − 1.25·22-s + 0.684·23-s − 0.152·24-s + 0.200·25-s + 0.568·26-s − 0.784·27-s + 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(2.157750260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157750260\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| 5 | \( 1 + 3.05e10T \) |
| good | 3 | \( 1 - 1.07e7T + 6.17e14T^{2} \) |
| 7 | \( 1 - 2.26e13T + 1.57e26T^{2} \) |
| 11 | \( 1 - 2.45e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 1.48e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 1.16e19T + 1.39e38T^{2} \) |
| 19 | \( 1 - 3.78e19T + 4.37e39T^{2} \) |
| 23 | \( 1 - 8.75e20T + 1.63e42T^{2} \) |
| 29 | \( 1 + 8.57e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 2.14e22T + 1.70e46T^{2} \) |
| 37 | \( 1 + 2.44e24T + 4.11e48T^{2} \) |
| 41 | \( 1 - 1.07e25T + 9.91e49T^{2} \) |
| 43 | \( 1 - 6.78e24T + 4.34e50T^{2} \) |
| 47 | \( 1 + 1.41e26T + 6.83e51T^{2} \) |
| 53 | \( 1 - 5.25e26T + 2.83e53T^{2} \) |
| 59 | \( 1 - 2.03e27T + 7.87e54T^{2} \) |
| 61 | \( 1 - 6.56e27T + 2.21e55T^{2} \) |
| 67 | \( 1 - 2.16e28T + 4.05e56T^{2} \) |
| 71 | \( 1 - 2.83e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 2.94e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 2.36e29T + 6.70e58T^{2} \) |
| 83 | \( 1 - 7.97e29T + 3.10e59T^{2} \) |
| 89 | \( 1 - 1.17e30T + 2.69e60T^{2} \) |
| 97 | \( 1 + 3.02e30T + 3.88e61T^{2} \) |
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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
−14.43001746334773705877343598807, −11.87716856148701995697642440913, −11.24952737001225955935701874864, −9.344886200642484944198761888823, −8.285495702833957877442910728068, −7.27982344695646936241333726631, −5.30216774150479921347713117739, −3.65887322428048848758089559006, −1.98951857919564664797491242082, −0.912107971976522102988485187780,
0.912107971976522102988485187780, 1.98951857919564664797491242082, 3.65887322428048848758089559006, 5.30216774150479921347713117739, 7.27982344695646936241333726631, 8.285495702833957877442910728068, 9.344886200642484944198761888823, 11.24952737001225955935701874864, 11.87716856148701995697642440913, 14.43001746334773705877343598807
