| L(s) = 1 | − 26.8·2-s + 81·3-s + 209.·4-s − 2.74e3·5-s − 2.17e3·6-s − 8.87e3·7-s + 8.12e3·8-s + 6.56e3·9-s + 7.36e4·10-s − 6.17e4·11-s + 1.69e4·12-s − 9.34e4·13-s + 2.38e5·14-s − 2.22e5·15-s − 3.25e5·16-s + 5.40e5·17-s − 1.76e5·18-s + 5.91e5·19-s − 5.74e5·20-s − 7.19e5·21-s + 1.65e6·22-s + 5.71e5·23-s + 6.58e5·24-s + 5.56e6·25-s + 2.50e6·26-s + 5.31e5·27-s − 1.85e6·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.577·3-s + 0.408·4-s − 1.96·5-s − 0.685·6-s − 1.39·7-s + 0.701·8-s + 0.333·9-s + 2.32·10-s − 1.27·11-s + 0.236·12-s − 0.907·13-s + 1.65·14-s − 1.13·15-s − 1.24·16-s + 1.56·17-s − 0.395·18-s + 1.04·19-s − 0.802·20-s − 0.806·21-s + 1.50·22-s + 0.425·23-s + 0.405·24-s + 2.85·25-s + 1.07·26-s + 0.192·27-s − 0.571·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 26.8T + 512T^{2} \) |
| 5 | \( 1 + 2.74e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.34e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.40e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.71e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.71e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.91e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.70e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.11e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 5.31e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.29e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 7.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.55e8T + 7.60e17T^{2} \) |
| 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
−10.13960461666925492579742903707, −9.545555902510325292186301380105, −8.336735346139023023819145185067, −7.61373841732540258849398634977, −7.17508360670281534929466252785, −5.02035697316529452829356784736, −3.62516427876987312187203470181, −2.85743321598984424921079458211, −0.78487227138101284743329803731, 0,
0.78487227138101284743329803731, 2.85743321598984424921079458211, 3.62516427876987312187203470181, 5.02035697316529452829356784736, 7.17508360670281534929466252785, 7.61373841732540258849398634977, 8.336735346139023023819145185067, 9.545555902510325292186301380105, 10.13960461666925492579742903707
