L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s + 6·11-s − 3·13-s − 4·16-s + 4·17-s + 19-s − 2·20-s + 12·22-s + 4·23-s + 25-s − 6·26-s + 8·29-s + 31-s − 8·32-s + 8·34-s + 7·37-s + 2·38-s + 6·41-s + 43-s + 12·44-s + 8·46-s − 2·47-s + 2·50-s − 6·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s + 1.80·11-s − 0.832·13-s − 16-s + 0.970·17-s + 0.229·19-s − 0.447·20-s + 2.55·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s + 1.48·29-s + 0.179·31-s − 1.41·32-s + 1.37·34-s + 1.15·37-s + 0.324·38-s + 0.937·41-s + 0.152·43-s + 1.80·44-s + 1.17·46-s − 0.291·47-s + 0.282·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.788284837\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.788284837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.177531078221159546886211566123, −8.162409396722371604713981977473, −7.19123664331710969417701675281, −6.55165825007787800789384133862, −5.82038228243474919000530168044, −4.82618716878877905203974067275, −4.28013905942012292848004574594, −3.43832873946187846440701924291, −2.66465413795447853148537423791, −1.10671322270974026757009572862,
1.10671322270974026757009572862, 2.66465413795447853148537423791, 3.43832873946187846440701924291, 4.28013905942012292848004574594, 4.82618716878877905203974067275, 5.82038228243474919000530168044, 6.55165825007787800789384133862, 7.19123664331710969417701675281, 8.162409396722371604713981977473, 9.177531078221159546886211566123