| L(s) = 1 | + 0.347·2-s + 3-s − 1.87·4-s − 1.34·5-s + 0.347·6-s − 0.652·7-s − 1.34·8-s + 9-s − 0.467·10-s + 3.94·11-s − 1.87·12-s − 1.69·13-s − 0.226·14-s − 1.34·15-s + 3.29·16-s − 3.83·17-s + 0.347·18-s + 2.53·20-s − 0.652·21-s + 1.36·22-s − 3.63·23-s − 1.34·24-s − 3.18·25-s − 0.588·26-s + 27-s + 1.22·28-s − 10.5·29-s + ⋯ |
| L(s) = 1 | + 0.245·2-s + 0.577·3-s − 0.939·4-s − 0.602·5-s + 0.141·6-s − 0.246·7-s − 0.476·8-s + 0.333·9-s − 0.147·10-s + 1.18·11-s − 0.542·12-s − 0.469·13-s − 0.0605·14-s − 0.347·15-s + 0.822·16-s − 0.930·17-s + 0.0818·18-s + 0.566·20-s − 0.142·21-s + 0.291·22-s − 0.758·23-s − 0.275·24-s − 0.636·25-s − 0.115·26-s + 0.192·27-s + 0.231·28-s − 1.95·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 - 2.94T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 - 4.31T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−9.423707974779152735340144628600, −8.715343159726752698457853086915, −7.932988318545102605537739839666, −7.04525855571811039392869532583, −6.03299243916312091609718484813, −4.90029569959468801424097765184, −3.92782003901918499052843945430, −3.54058177996859447645360824686, −1.90681417130665009428804541336, 0,
1.90681417130665009428804541336, 3.54058177996859447645360824686, 3.92782003901918499052843945430, 4.90029569959468801424097765184, 6.03299243916312091609718484813, 7.04525855571811039392869532583, 7.932988318545102605537739839666, 8.715343159726752698457853086915, 9.423707974779152735340144628600
