| L(s) = 1 | − 29.2·2-s + 77.9·3-s + 599·4-s + 625·5-s − 2.27e3·6-s − 1.00e4·8-s − 481.·9-s − 1.82e4·10-s + 2.54e4·11-s + 4.67e4·12-s + 2.22e4·13-s + 4.87e4·15-s + 1.39e5·16-s + 1.40e4·18-s + 1.30e5·19-s + 3.74e5·20-s − 7.43e5·22-s − 7.82e5·24-s + 3.90e5·25-s − 6.49e5·26-s − 5.49e5·27-s − 1.42e6·30-s − 1.52e6·32-s + 1.98e6·33-s − 2.88e5·36-s − 1.88e6·37-s − 3.81e6·38-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 0.962·3-s + 2.33·4-s + 5-s − 1.75·6-s − 2.44·8-s − 0.0733·9-s − 1.82·10-s + 1.73·11-s + 2.25·12-s + 0.778·13-s + 0.962·15-s + 2.13·16-s + 0.133·18-s + 19-s + 2.33·20-s − 3.17·22-s − 2.35·24-s + 25-s − 1.42·26-s − 1.03·27-s − 1.75·30-s − 1.45·32-s + 1.67·33-s − 0.171·36-s − 1.00·37-s − 1.82·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.732031683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732031683\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 + 29.2T + 256T^{2} \) |
| 3 | \( 1 - 77.9T + 6.56e3T^{2} \) |
| 7 | \( 1 - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.54e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 2.22e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.97e9T^{2} \) |
| 23 | \( 1 - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.88e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.16e13T^{2} \) |
| 47 | \( 1 - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.40e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.32e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 4.01e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.45e14T^{2} \) |
| 73 | \( 1 - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.66e8T + 7.83e15T^{2} \) |
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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
−11.96900630041085929675338776446, −10.90816445289999546413956588704, −9.661722703716370405830380578792, −9.132863576895389497247545246430, −8.405495497754468146662675232937, −7.07321673249459922310926133138, −6.03105684952951395849527094346, −3.31641366754077334968475562665, −1.95674308547374988458408806991, −1.06220855300898680125899061553,
1.06220855300898680125899061553, 1.95674308547374988458408806991, 3.31641366754077334968475562665, 6.03105684952951395849527094346, 7.07321673249459922310926133138, 8.405495497754468146662675232937, 9.132863576895389497247545246430, 9.661722703716370405830380578792, 10.90816445289999546413956588704, 11.96900630041085929675338776446
