| L(s) = 1 | + 2.69·2-s − 3-s + 5.23·4-s − 3.39·5-s − 2.69·6-s + 3.55·7-s + 8.71·8-s + 9-s − 9.12·10-s + 3.97·11-s − 5.23·12-s + 13-s + 9.57·14-s + 3.39·15-s + 12.9·16-s − 5.69·17-s + 2.69·18-s − 3.19·19-s − 17.7·20-s − 3.55·21-s + 10.6·22-s + 4.70·23-s − 8.71·24-s + 6.49·25-s + 2.69·26-s − 27-s + 18.6·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.61·4-s − 1.51·5-s − 1.09·6-s + 1.34·7-s + 3.08·8-s + 0.333·9-s − 2.88·10-s + 1.19·11-s − 1.51·12-s + 0.277·13-s + 2.55·14-s + 0.875·15-s + 3.24·16-s − 1.38·17-s + 0.634·18-s − 0.732·19-s − 3.97·20-s − 0.776·21-s + 2.27·22-s + 0.981·23-s − 1.77·24-s + 1.29·25-s + 0.527·26-s − 0.192·27-s + 3.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.513261643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.513261643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 6.05T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 2.79T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 9.35T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.253261089493115356373578235155, −7.30737618362928527584654129270, −6.76905066032857596468060286023, −6.22628846608848718154210062480, −5.04206689921684264405724170933, −4.59527374105784617472495169624, −4.16282851578012103481679865714, −3.42166353250805979118557137991, −2.23210601427318104162047514129, −1.13183107888023680539055164761,
1.13183107888023680539055164761, 2.23210601427318104162047514129, 3.42166353250805979118557137991, 4.16282851578012103481679865714, 4.59527374105784617472495169624, 5.04206689921684264405724170933, 6.22628846608848718154210062480, 6.76905066032857596468060286023, 7.30737618362928527584654129270, 8.253261089493115356373578235155
