| L(s) = 1 | + 0.326·2-s + 1.56·3-s − 1.89·4-s − 5-s + 0.511·6-s − 1.26·8-s − 0.539·9-s − 0.326·10-s + 4.81·11-s − 2.97·12-s − 4.41·13-s − 1.56·15-s + 3.37·16-s − 1.70·17-s − 0.175·18-s − 2.34·19-s + 1.89·20-s + 1.57·22-s + 5.27·23-s − 1.99·24-s + 25-s − 1.44·26-s − 5.55·27-s + 9.43·29-s − 0.511·30-s − 1.79·31-s + 3.63·32-s + ⋯ |
| L(s) = 1 | + 0.230·2-s + 0.905·3-s − 0.946·4-s − 0.447·5-s + 0.208·6-s − 0.448·8-s − 0.179·9-s − 0.103·10-s + 1.45·11-s − 0.857·12-s − 1.22·13-s − 0.404·15-s + 0.843·16-s − 0.413·17-s − 0.0414·18-s − 0.537·19-s + 0.423·20-s + 0.334·22-s + 1.10·23-s − 0.406·24-s + 0.200·25-s − 0.282·26-s − 1.06·27-s + 1.75·29-s − 0.0933·30-s − 0.322·31-s + 0.643·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.326T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 8.00T + 67T^{2} \) |
| 71 | \( 1 - 5.31T + 71T^{2} \) |
| 73 | \( 1 + 3.64T + 73T^{2} \) |
| 79 | \( 1 + 6.03T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.38363786520565606837315513888, −6.86101061375580945459724965826, −6.01128324869858009710180672884, −5.06186457519030085683821539990, −4.49260015108917864008254506383, −3.85765097956370951956410843597, −3.15347565770778813211757484580, −2.44370830178230666774074693214, −1.21310855965747987237397694087, 0,
1.21310855965747987237397694087, 2.44370830178230666774074693214, 3.15347565770778813211757484580, 3.85765097956370951956410843597, 4.49260015108917864008254506383, 5.06186457519030085683821539990, 6.01128324869858009710180672884, 6.86101061375580945459724965826, 7.38363786520565606837315513888
