| L(s) = 1 | + 0.671·3-s + 5-s + 0.305·7-s − 2.54·9-s − 1.85·11-s + 4.78·13-s + 0.671·15-s − 2.54·17-s − 8.13·19-s + 0.204·21-s − 6.67·23-s + 25-s − 3.72·27-s + 0.374·29-s + 4.02·31-s − 1.24·33-s + 0.305·35-s − 37-s + 3.21·39-s + 3.06·41-s − 11.5·43-s − 2.54·45-s − 4.13·47-s − 6.90·49-s − 1.71·51-s − 2.51·53-s − 1.85·55-s + ⋯ |
| L(s) = 1 | + 0.387·3-s + 0.447·5-s + 0.115·7-s − 0.849·9-s − 0.560·11-s + 1.32·13-s + 0.173·15-s − 0.617·17-s − 1.86·19-s + 0.0447·21-s − 1.39·23-s + 0.200·25-s − 0.717·27-s + 0.0696·29-s + 0.722·31-s − 0.217·33-s + 0.0515·35-s − 0.164·37-s + 0.514·39-s + 0.478·41-s − 1.75·43-s − 0.379·45-s − 0.603·47-s − 0.986·49-s − 0.239·51-s − 0.345·53-s − 0.250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 0.671T + 3T^{2} \) |
| 7 | \( 1 - 0.305T + 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 8.13T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 0.374T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 - 8.57T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 8.41T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.339751358516737809931634216190, −8.010801354655118991936239045137, −6.53660704898464250708581402133, −6.26853810511527174099124418673, −5.35618980681432427587515435272, −4.37468426671051995729586432162, −3.52263289477318909486113188644, −2.50692114123233260066549481481, −1.74521948888927097265488773365, 0,
1.74521948888927097265488773365, 2.50692114123233260066549481481, 3.52263289477318909486113188644, 4.37468426671051995729586432162, 5.35618980681432427587515435272, 6.26853810511527174099124418673, 6.53660704898464250708581402133, 8.010801354655118991936239045137, 8.339751358516737809931634216190
