| L(s) = 1 | − 3·3-s + 3.18·5-s − 23.9·7-s + 9·9-s + 6.74·11-s + 13·13-s − 9.55·15-s + 104.·17-s + 137.·19-s + 71.8·21-s − 110.·23-s − 114.·25-s − 27·27-s − 57.6·29-s − 319.·31-s − 20.2·33-s − 76.2·35-s − 2.88·37-s − 39·39-s + 319.·41-s − 344.·43-s + 28.6·45-s + 439.·47-s + 229.·49-s − 312.·51-s − 97.2·53-s + 21.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.285·5-s − 1.29·7-s + 0.333·9-s + 0.184·11-s + 0.277·13-s − 0.164·15-s + 1.48·17-s + 1.66·19-s + 0.746·21-s − 0.999·23-s − 0.918·25-s − 0.192·27-s − 0.368·29-s − 1.85·31-s − 0.106·33-s − 0.368·35-s − 0.0128·37-s − 0.160·39-s + 1.21·41-s − 1.22·43-s + 0.0950·45-s + 1.36·47-s + 0.670·49-s − 0.858·51-s − 0.252·53-s + 0.0527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 3.18T + 125T^{2} \) |
| 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 - 6.74T + 1.33e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.88T + 5.06e4T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 344.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 666.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 828.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 734.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 153.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 569.T + 9.12e5T^{2} \) |
| 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
−9.726269190880028284519500884620, −9.293218749784898745965609821990, −7.79027363598270318609980440252, −7.07639955972183165464598907718, −5.84049468002622530892331759958, −5.61335225456098238290590488733, −3.94550600104159854951898584534, −3.10057993768717089241298325233, −1.42187229870014565175242963864, 0,
1.42187229870014565175242963864, 3.10057993768717089241298325233, 3.94550600104159854951898584534, 5.61335225456098238290590488733, 5.84049468002622530892331759958, 7.07639955972183165464598907718, 7.79027363598270318609980440252, 9.293218749784898745965609821990, 9.726269190880028284519500884620
