| L(s) = 1 | + 3-s + 5-s − 3.12·7-s + 9-s − 1.51·11-s − 3.60·13-s + 15-s + 5.28·17-s − 1.51·19-s − 3.12·21-s + 1.76·23-s + 25-s + 27-s + 1.60·29-s − 31-s − 1.51·33-s − 3.12·35-s + 4.57·37-s − 3.60·39-s + 0.249·41-s − 10.7·43-s + 45-s + 6.49·47-s + 2.76·49-s + 5.28·51-s + 3.45·53-s − 1.51·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.18·7-s + 0.333·9-s − 0.456·11-s − 1.00·13-s + 0.258·15-s + 1.28·17-s − 0.347·19-s − 0.681·21-s + 0.368·23-s + 0.200·25-s + 0.192·27-s + 0.298·29-s − 0.179·31-s − 0.263·33-s − 0.528·35-s + 0.752·37-s − 0.578·39-s + 0.0390·41-s − 1.64·43-s + 0.149·45-s + 0.948·47-s + 0.394·49-s + 0.739·51-s + 0.474·53-s − 0.204·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 - 0.249T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 - 3.45T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 5.45T + 79T^{2} \) |
| 83 | \( 1 + 6.96T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−7.56881483731257947875579700469, −6.89408179629441774476823954790, −6.21567835083395500670068353012, −5.46297143393809925112500459066, −4.74571768116410202645070113601, −3.76054776909548256797670099251, −2.97658184072610919353829171392, −2.52541536767232578616575298138, −1.35366246390816957199490799969, 0,
1.35366246390816957199490799969, 2.52541536767232578616575298138, 2.97658184072610919353829171392, 3.76054776909548256797670099251, 4.74571768116410202645070113601, 5.46297143393809925112500459066, 6.21567835083395500670068353012, 6.89408179629441774476823954790, 7.56881483731257947875579700469
