| L(s) = 1 | + 128.·3-s − 1.33e3·5-s + 2.40e3·7-s − 3.18e3·9-s − 2.22e4·11-s + 1.85e5·13-s − 1.71e5·15-s + 3.51e5·17-s + 1.10e6·19-s + 3.08e5·21-s − 1.99e6·23-s − 1.75e5·25-s − 2.93e6·27-s + 3.18e6·29-s + 7.42e6·31-s − 2.86e6·33-s − 3.20e6·35-s + 1.11e7·37-s + 2.37e7·39-s + 2.22e7·41-s + 1.98e7·43-s + 4.24e6·45-s − 5.69e5·47-s + 5.76e6·49-s + 4.51e7·51-s − 6.76e7·53-s + 2.97e7·55-s + ⋯ |
| L(s) = 1 | + 0.915·3-s − 0.954·5-s + 0.377·7-s − 0.161·9-s − 0.458·11-s + 1.79·13-s − 0.873·15-s + 1.02·17-s + 1.95·19-s + 0.346·21-s − 1.48·23-s − 0.0897·25-s − 1.06·27-s + 0.835·29-s + 1.44·31-s − 0.420·33-s − 0.360·35-s + 0.978·37-s + 1.64·39-s + 1.23·41-s + 0.886·43-s + 0.154·45-s − 0.0170·47-s + 0.142·49-s + 0.935·51-s − 1.17·53-s + 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.506412936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506412936\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 128.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.33e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.22e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.85e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.51e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.10e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.99e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.11e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.22e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.98e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.69e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.76e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.40e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.82e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.61e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.80e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.34e8T + 7.60e17T^{2} \) |
| 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
−13.67941605621271854674270687685, −12.06761697639589523946896111265, −11.17482019418171147203415852741, −9.651738908862816910736983467061, −8.174024261304149969154381203850, −7.85341896366426524742960930714, −5.82212738134335334978610819453, −3.98816829751607567870208128470, −2.95104569017411972192904587669, −1.02437846945909118016048495455,
1.02437846945909118016048495455, 2.95104569017411972192904587669, 3.98816829751607567870208128470, 5.82212738134335334978610819453, 7.85341896366426524742960930714, 8.174024261304149969154381203850, 9.651738908862816910736983467061, 11.17482019418171147203415852741, 12.06761697639589523946896111265, 13.67941605621271854674270687685
