L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 4·19-s − 4·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 4·31-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s − 10·43-s + 45-s − 6·47-s − 3·49-s + 12·51-s + 6·53-s − 8·57-s − 12·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.90965367907189, −14.62061111353371, −14.04794149368439, −13.51676254092497, −13.11802471903221, −12.72430738732027, −12.01966848761303, −11.48032148380751, −10.67796753468791, −10.20308820178249, −9.776020543409207, −9.169680301534925, −8.648922499328547, −8.345336398850424, −7.626883217511953, −6.997632899166327, −6.434791849274886, −5.871483014492509, −5.181866785898032, −4.513646985194083, −3.523213216003887, −3.265622796090793, −2.764955856684508, −1.843163717265040, −1.241692441308918, 0,
1.241692441308918, 1.843163717265040, 2.764955856684508, 3.265622796090793, 3.523213216003887, 4.513646985194083, 5.181866785898032, 5.871483014492509, 6.434791849274886, 6.997632899166327, 7.626883217511953, 8.345336398850424, 8.648922499328547, 9.169680301534925, 9.776020543409207, 10.20308820178249, 10.67796753468791, 11.48032148380751, 12.01966848761303, 12.72430738732027, 13.11802471903221, 13.51676254092497, 14.04794149368439, 14.62061111353371, 14.90965367907189