| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3.23·7-s + 8-s + 9-s + 4.47·11-s − 12-s − 13-s − 3.23·14-s + 16-s − 7.23·17-s + 18-s − 2.76·19-s + 3.23·21-s + 4.47·22-s − 2.76·23-s − 24-s − 26-s − 27-s − 3.23·28-s − 3.70·29-s − 4·31-s + 32-s − 4.47·33-s − 7.23·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.22·7-s + 0.353·8-s + 0.333·9-s + 1.34·11-s − 0.288·12-s − 0.277·13-s − 0.864·14-s + 0.250·16-s − 1.75·17-s + 0.235·18-s − 0.634·19-s + 0.706·21-s + 0.953·22-s − 0.576·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.611·28-s − 0.688·29-s − 0.718·31-s + 0.176·32-s − 0.778·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−9.119788551672627996137290449595, −7.79421724753621842560668008589, −6.74554511709545220295781459014, −6.43717490653557859344038888203, −5.78424952265621951803630672333, −4.47134180665468607589785062244, −4.05826633733612975549111607825, −2.93863753798985817103168080001, −1.76013233051048305859997525648, 0,
1.76013233051048305859997525648, 2.93863753798985817103168080001, 4.05826633733612975549111607825, 4.47134180665468607589785062244, 5.78424952265621951803630672333, 6.43717490653557859344038888203, 6.74554511709545220295781459014, 7.79421724753621842560668008589, 9.119788551672627996137290449595
