| L(s) = 1 | + 1.13·2-s + 3.45·3-s − 6.70·4-s + 5·5-s + 3.94·6-s + 1.89·7-s − 16.7·8-s − 15.0·9-s + 5.69·10-s + 11·11-s − 23.1·12-s − 13.9·13-s + 2.15·14-s + 17.2·15-s + 34.5·16-s + 98.8·17-s − 17.1·18-s + 19·19-s − 33.5·20-s + 6.53·21-s + 12.5·22-s − 158.·23-s − 57.9·24-s + 25·25-s − 15.9·26-s − 145.·27-s − 12.6·28-s + ⋯ |
| L(s) = 1 | + 0.403·2-s + 0.665·3-s − 0.837·4-s + 0.447·5-s + 0.268·6-s + 0.102·7-s − 0.740·8-s − 0.557·9-s + 0.180·10-s + 0.301·11-s − 0.557·12-s − 0.298·13-s + 0.0411·14-s + 0.297·15-s + 0.539·16-s + 1.40·17-s − 0.224·18-s + 0.229·19-s − 0.374·20-s + 0.0679·21-s + 0.121·22-s − 1.43·23-s − 0.492·24-s + 0.200·25-s − 0.120·26-s − 1.03·27-s − 0.0855·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 1.13T + 8T^{2} \) |
| 3 | \( 1 - 3.45T + 27T^{2} \) |
| 7 | \( 1 - 1.89T + 343T^{2} \) |
| 13 | \( 1 + 13.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.58T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 13.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 433.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 464.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 673.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 22.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 404.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 64.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 895.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 607.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
| 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
−9.248945312538190711075239020233, −8.243597835033796937506394436193, −7.79824416644610960390648306065, −6.33063882980100352931332523082, −5.59899738621490887416097013729, −4.75118544605065048847278388492, −3.62637831398809220881270786051, −2.93992507760170753226074102514, −1.56630800513363904968317720447, 0,
1.56630800513363904968317720447, 2.93992507760170753226074102514, 3.62637831398809220881270786051, 4.75118544605065048847278388492, 5.59899738621490887416097013729, 6.33063882980100352931332523082, 7.79824416644610960390648306065, 8.243597835033796937506394436193, 9.248945312538190711075239020233
