| L(s) = 1 | + 1.41·3-s + 5-s − 2.82·7-s − 0.999·9-s + 4.82·11-s − 0.585·13-s + 1.41·15-s − 0.828·17-s − 19-s − 4.00·21-s − 4·23-s + 25-s − 5.65·27-s − 4.82·29-s + 6.82·33-s − 2.82·35-s − 1.75·37-s − 0.828·39-s + 4.82·41-s − 2.82·43-s − 0.999·45-s − 8.48·47-s + 1.00·49-s − 1.17·51-s + 1.07·53-s + 4.82·55-s − 1.41·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.447·5-s − 1.06·7-s − 0.333·9-s + 1.45·11-s − 0.162·13-s + 0.365·15-s − 0.200·17-s − 0.229·19-s − 0.872·21-s − 0.834·23-s + 0.200·25-s − 1.08·27-s − 0.896·29-s + 1.18·33-s − 0.478·35-s − 0.288·37-s − 0.132·39-s + 0.754·41-s − 0.431·43-s − 0.149·45-s − 1.23·47-s + 0.142·49-s − 0.164·51-s + 0.147·53-s + 0.651·55-s − 0.187·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 1.07T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
| 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
−7.76760565092107050697203067396, −6.96377445862106700441733988411, −6.23975023577786343083606399407, −5.87689702938886043946719670477, −4.69432521342562656719993507575, −3.75934419873765209491993547512, −3.30192175563577018740494259590, −2.37460344817134277853307979169, −1.54349588144185699979579775229, 0,
1.54349588144185699979579775229, 2.37460344817134277853307979169, 3.30192175563577018740494259590, 3.75934419873765209491993547512, 4.69432521342562656719993507575, 5.87689702938886043946719670477, 6.23975023577786343083606399407, 6.96377445862106700441733988411, 7.76760565092107050697203067396
