| L(s) = 1 | + 1.44·2-s + 1.62·3-s + 0.100·4-s + 5-s + 2.35·6-s − 7-s − 2.75·8-s − 0.356·9-s + 1.44·10-s + 0.572·11-s + 0.163·12-s − 0.626·13-s − 1.44·14-s + 1.62·15-s − 4.19·16-s − 5.92·17-s − 0.516·18-s + 6.78·19-s + 0.100·20-s − 1.62·21-s + 0.829·22-s + 7.65·23-s − 4.47·24-s + 25-s − 0.907·26-s − 5.45·27-s − 0.100·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.938·3-s + 0.0504·4-s + 0.447·5-s + 0.962·6-s − 0.377·7-s − 0.973·8-s − 0.118·9-s + 0.458·10-s + 0.172·11-s + 0.0473·12-s − 0.173·13-s − 0.387·14-s + 0.419·15-s − 1.04·16-s − 1.43·17-s − 0.121·18-s + 1.55·19-s + 0.0225·20-s − 0.354·21-s + 0.176·22-s + 1.59·23-s − 0.913·24-s + 0.200·25-s − 0.178·26-s − 1.05·27-s − 0.0190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 11 | \( 1 - 0.572T + 11T^{2} \) |
| 13 | \( 1 + 0.626T + 13T^{2} \) |
| 17 | \( 1 + 5.92T + 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 + 0.607T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 0.434T + 97T^{2} \) |
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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
−7.37583391209309499419540275740, −6.65033288504856215237896921805, −6.07204273403475502500568304857, −5.05124650931662381032192456035, −4.88990157213943885289024990007, −3.64442386536073734523573911567, −3.24943663299866623588382448495, −2.61885243014413110220247355951, −1.62251138303809900417057295462, 0,
1.62251138303809900417057295462, 2.61885243014413110220247355951, 3.24943663299866623588382448495, 3.64442386536073734523573911567, 4.88990157213943885289024990007, 5.05124650931662381032192456035, 6.07204273403475502500568304857, 6.65033288504856215237896921805, 7.37583391209309499419540275740
