| L(s) = 1 | − 2.24·2-s + 2.50·3-s + 3.05·4-s + 1.29·5-s − 5.62·6-s + 3.89·7-s − 2.36·8-s + 3.26·9-s − 2.91·10-s + 11-s + 7.63·12-s + 5.45·13-s − 8.75·14-s + 3.25·15-s − 0.793·16-s − 17-s − 7.33·18-s + 5.05·19-s + 3.96·20-s + 9.75·21-s − 2.24·22-s + 6.25·23-s − 5.91·24-s − 3.31·25-s − 12.2·26-s + 0.663·27-s + 11.8·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.44·3-s + 1.52·4-s + 0.580·5-s − 2.29·6-s + 1.47·7-s − 0.835·8-s + 1.08·9-s − 0.922·10-s + 0.301·11-s + 2.20·12-s + 1.51·13-s − 2.34·14-s + 0.839·15-s − 0.198·16-s − 0.242·17-s − 1.72·18-s + 1.15·19-s + 0.885·20-s + 2.12·21-s − 0.479·22-s + 1.30·23-s − 1.20·24-s − 0.662·25-s − 2.40·26-s + 0.127·27-s + 2.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.707086830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707086830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
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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
−8.191892910979238452367565241979, −7.39502042151645158278527248600, −7.01669085391173059718336540423, −5.90820324534055847219409693587, −5.08105229234887635484456621628, −4.04192334742226832563963609178, −3.22847607559766727571193685758, −2.26315907863157275260043459262, −1.59854747936950249723117595648, −1.11373474218063785514946250136,
1.11373474218063785514946250136, 1.59854747936950249723117595648, 2.26315907863157275260043459262, 3.22847607559766727571193685758, 4.04192334742226832563963609178, 5.08105229234887635484456621628, 5.90820324534055847219409693587, 7.01669085391173059718336540423, 7.39502042151645158278527248600, 8.191892910979238452367565241979
