| L(s) = 1 | − 32.9·2-s + 81·3-s + 575.·4-s − 958.·5-s − 2.67e3·6-s − 5.51e3·7-s − 2.10e3·8-s + 6.56e3·9-s + 3.16e4·10-s + 2.93e4·11-s + 4.66e4·12-s + 1.37e5·13-s + 1.81e5·14-s − 7.76e4·15-s − 2.25e5·16-s − 3.39e5·17-s − 2.16e5·18-s + 8.62e4·19-s − 5.51e5·20-s − 4.46e5·21-s − 9.66e5·22-s + 1.13e6·23-s − 1.70e5·24-s − 1.03e6·25-s − 4.55e6·26-s + 5.31e5·27-s − 3.17e6·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.12·4-s − 0.685·5-s − 0.841·6-s − 0.867·7-s − 0.181·8-s + 0.333·9-s + 0.999·10-s + 0.603·11-s + 0.649·12-s + 1.33·13-s + 1.26·14-s − 0.395·15-s − 0.860·16-s − 0.986·17-s − 0.485·18-s + 0.151·19-s − 0.771·20-s − 0.501·21-s − 0.880·22-s + 0.845·23-s − 0.104·24-s − 0.529·25-s − 1.95·26-s + 0.192·27-s − 0.975·28-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
| \(L(5)\) |
\(\approx\) |
\(0.8279116790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8279116790\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 32.9T + 512T^{2} \) |
| 5 | \( 1 + 958.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.51e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.37e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.62e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 9.19e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.08e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.63e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.33e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.15e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.09e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.97e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.19e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.25e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.70e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.88e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.09e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.82529934224543496254633241232, −9.691297790823039023323605853666, −8.956805834561907865164004150754, −8.276121756272080035225111948369, −7.20243923875020173426246894174, −6.33979705365806666746617346564, −4.25199741511256758931590161473, −3.18130669348122895470669012603, −1.68910134600434932410759260745, −0.55628994279862697143417450978,
0.55628994279862697143417450978, 1.68910134600434932410759260745, 3.18130669348122895470669012603, 4.25199741511256758931590161473, 6.33979705365806666746617346564, 7.20243923875020173426246894174, 8.276121756272080035225111948369, 8.956805834561907865164004150754, 9.691297790823039023323605853666, 10.82529934224543496254633241232
