| L(s) = 1 | − 1.31·2-s + 2.24·3-s − 0.266·4-s − 5-s − 2.95·6-s + 2.74·7-s + 2.98·8-s + 2.05·9-s + 1.31·10-s − 11-s − 0.598·12-s − 0.823·13-s − 3.61·14-s − 2.24·15-s − 3.39·16-s + 4.43·17-s − 2.70·18-s + 19-s + 0.266·20-s + 6.16·21-s + 1.31·22-s + 5.39·23-s + 6.70·24-s + 25-s + 1.08·26-s − 2.13·27-s − 0.730·28-s + ⋯ |
| L(s) = 1 | − 0.931·2-s + 1.29·3-s − 0.133·4-s − 0.447·5-s − 1.20·6-s + 1.03·7-s + 1.05·8-s + 0.683·9-s + 0.416·10-s − 0.301·11-s − 0.172·12-s − 0.228·13-s − 0.965·14-s − 0.580·15-s − 0.849·16-s + 1.07·17-s − 0.636·18-s + 0.229·19-s + 0.0595·20-s + 1.34·21-s + 0.280·22-s + 1.12·23-s + 1.36·24-s + 0.200·25-s + 0.212·26-s − 0.410·27-s − 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470470170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470470170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 13 | \( 1 + 0.823T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 + 0.797T + 73T^{2} \) |
| 79 | \( 1 + 0.962T + 79T^{2} \) |
| 83 | \( 1 - 0.110T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.630981410076687507454030891312, −8.922775421674158000046333357080, −8.384026349112014197861436160854, −7.61929480985784697982796205137, −7.33857365352876099057943069978, −5.46271389151439024789407612961, −4.53521329678593726368004697209, −3.52578029070233837747278145838, −2.35279690439566532796855785869, −1.09655231698416078015532763285,
1.09655231698416078015532763285, 2.35279690439566532796855785869, 3.52578029070233837747278145838, 4.53521329678593726368004697209, 5.46271389151439024789407612961, 7.33857365352876099057943069978, 7.61929480985784697982796205137, 8.384026349112014197861436160854, 8.922775421674158000046333357080, 9.630981410076687507454030891312
