| L(s) = 1 | + 1.18·2-s − 0.602·4-s − 5-s − 5.08·7-s − 3.07·8-s − 1.18·10-s + 3.05·11-s + 13-s − 6.01·14-s − 2.43·16-s − 1.22·17-s + 3.82·19-s + 0.602·20-s + 3.61·22-s + 6.12·23-s + 25-s + 1.18·26-s + 3.06·28-s + 9.62·29-s − 8.15·31-s + 3.27·32-s − 1.44·34-s + 5.08·35-s + 10.7·37-s + 4.52·38-s + 3.07·40-s + 1.43·41-s + ⋯ |
| L(s) = 1 | + 0.835·2-s − 0.301·4-s − 0.447·5-s − 1.92·7-s − 1.08·8-s − 0.373·10-s + 0.921·11-s + 0.277·13-s − 1.60·14-s − 0.608·16-s − 0.297·17-s + 0.878·19-s + 0.134·20-s + 0.769·22-s + 1.27·23-s + 0.200·25-s + 0.231·26-s + 0.579·28-s + 1.78·29-s − 1.46·31-s + 0.579·32-s − 0.248·34-s + 0.859·35-s + 1.76·37-s + 0.734·38-s + 0.486·40-s + 0.223·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507196783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507196783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 - 3.10T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.69T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−9.302933391646706074458009415684, −8.797991575524619135883689654529, −7.54663679779998790611249779791, −6.49689770342569259205773081282, −6.27894840672569579450162561347, −5.12141876220799299615710714804, −4.20292672421444700062524981829, −3.38884338176972742163612814489, −2.91822131557416709626744833807, −0.73592956827733224895201950190,
0.73592956827733224895201950190, 2.91822131557416709626744833807, 3.38884338176972742163612814489, 4.20292672421444700062524981829, 5.12141876220799299615710714804, 6.27894840672569579450162561347, 6.49689770342569259205773081282, 7.54663679779998790611249779791, 8.797991575524619135883689654529, 9.302933391646706074458009415684
