| L(s) = 1 | + 2.61·2-s + 4.85·4-s − 2.23·5-s + 7.47·8-s − 5.85·10-s + 3·11-s − 13-s + 9.85·16-s − 1.47·17-s + 3·19-s − 10.8·20-s + 7.85·22-s + 8.23·23-s − 2.61·26-s − 4.47·29-s + 5·31-s + 10.8·32-s − 3.85·34-s + 4.70·37-s + 7.85·38-s − 16.7·40-s + 4.47·41-s − 8·43-s + 14.5·44-s + 21.5·46-s + 7.47·47-s − 4.85·52-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s − 0.999·5-s + 2.64·8-s − 1.85·10-s + 0.904·11-s − 0.277·13-s + 2.46·16-s − 0.357·17-s + 0.688·19-s − 2.42·20-s + 1.67·22-s + 1.71·23-s − 0.513·26-s − 0.830·29-s + 0.898·31-s + 1.91·32-s − 0.660·34-s + 0.774·37-s + 1.27·38-s − 2.64·40-s + 0.698·41-s − 1.21·43-s + 2.19·44-s + 3.17·46-s + 1.08·47-s − 0.673·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.240507242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.240507242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.69629757132774707611651376452, −7.17955611398549923853908406109, −6.62305805131218474681205173489, −5.81398570354841287006827766299, −5.04922240682503243801843755772, −4.40802129489538029928248668667, −3.79572697239028377233237807751, −3.16810263603139382652102846667, −2.31518502414556948623382678579, −1.05093587368298752338815604664,
1.05093587368298752338815604664, 2.31518502414556948623382678579, 3.16810263603139382652102846667, 3.79572697239028377233237807751, 4.40802129489538029928248668667, 5.04922240682503243801843755772, 5.81398570354841287006827766299, 6.62305805131218474681205173489, 7.17955611398549923853908406109, 7.69629757132774707611651376452
