| L(s) = 1 | + 5-s − 2.55·7-s + 2.72·11-s + 7.12·13-s − 0.924·17-s + 7.51·19-s + 23-s + 25-s + 2.38·29-s + 0.866·31-s − 2.55·35-s + 0.352·37-s − 4.34·41-s + 13.3·47-s − 0.495·49-s − 3.99·53-s + 2.72·55-s + 3.84·59-s − 9.14·61-s + 7.12·65-s − 3.15·67-s + 6.07·71-s − 11.3·73-s − 6.94·77-s − 12.0·79-s + 6.35·83-s − 0.924·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.963·7-s + 0.821·11-s + 1.97·13-s − 0.224·17-s + 1.72·19-s + 0.208·23-s + 0.200·25-s + 0.442·29-s + 0.155·31-s − 0.431·35-s + 0.0580·37-s − 0.677·41-s + 1.94·47-s − 0.0707·49-s − 0.548·53-s + 0.367·55-s + 0.500·59-s − 1.17·61-s + 0.883·65-s − 0.385·67-s + 0.721·71-s − 1.32·73-s − 0.791·77-s − 1.35·79-s + 0.697·83-s − 0.100·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.611833994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.611833994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 + 0.924T + 17T^{2} \) |
| 19 | \( 1 - 7.51T + 19T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 0.866T + 31T^{2} \) |
| 37 | \( 1 - 0.352T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 6.35T + 83T^{2} \) |
| 89 | \( 1 - 9.71T + 89T^{2} \) |
| 97 | \( 1 - 8.76T + 97T^{2} \) |
| 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
−7.74151651941721753376528805424, −6.99810823511478750310205297497, −6.28003771821467296351035427081, −5.98335889173842748682797743447, −5.12716579028903978428987705110, −4.09398215309837419589631421432, −3.45060982657215529869078400413, −2.86847756713323311857326493321, −1.56441827587576104151232641959, −0.870580025507061478568019292232,
0.870580025507061478568019292232, 1.56441827587576104151232641959, 2.86847756713323311857326493321, 3.45060982657215529869078400413, 4.09398215309837419589631421432, 5.12716579028903978428987705110, 5.98335889173842748682797743447, 6.28003771821467296351035427081, 6.99810823511478750310205297497, 7.74151651941721753376528805424
