| L(s) = 1 | + 2.09·2-s + 2.36·4-s + 3.20·5-s − 0.388·7-s + 0.771·8-s + 6.70·10-s − 11-s + 13-s − 0.811·14-s − 3.12·16-s + 5.09·17-s + 5.49·19-s + 7.60·20-s − 2.09·22-s − 2.51·23-s + 5.29·25-s + 2.09·26-s − 0.919·28-s + 1.17·29-s + 2.69·31-s − 8.07·32-s + 10.6·34-s − 1.24·35-s + 6.39·37-s + 11.4·38-s + 2.47·40-s − 10.6·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.18·4-s + 1.43·5-s − 0.146·7-s + 0.272·8-s + 2.12·10-s − 0.301·11-s + 0.277·13-s − 0.216·14-s − 0.781·16-s + 1.23·17-s + 1.26·19-s + 1.70·20-s − 0.445·22-s − 0.525·23-s + 1.05·25-s + 0.409·26-s − 0.173·28-s + 0.218·29-s + 0.484·31-s − 1.42·32-s + 1.82·34-s − 0.210·35-s + 1.05·37-s + 1.86·38-s + 0.391·40-s − 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.551719444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.551719444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 + 0.388T + 7T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 6.39T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 + 3.84T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + 0.946T + 73T^{2} \) |
| 79 | \( 1 + 6.41T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−9.897773379729608307424372762188, −9.007459294631222943886826554062, −7.85972354223119167113652207800, −6.76911679337005421840996319177, −5.99466855939207456948559653294, −5.47249389708547037026629559918, −4.76110851768389546138124064565, −3.47393966148247565845727871490, −2.76766456605592450130782415714, −1.55016855940194882066878486949,
1.55016855940194882066878486949, 2.76766456605592450130782415714, 3.47393966148247565845727871490, 4.76110851768389546138124064565, 5.47249389708547037026629559918, 5.99466855939207456948559653294, 6.76911679337005421840996319177, 7.85972354223119167113652207800, 9.007459294631222943886826554062, 9.897773379729608307424372762188
