| L(s) = 1 | − 5.38·2-s − 2.96·4-s + 0.456·5-s + 166.·7-s + 188.·8-s − 2.46·10-s + 65.6·11-s − 689.·13-s − 897.·14-s − 920.·16-s − 737.·17-s − 609.·19-s − 1.35·20-s − 353.·22-s − 1.31e3·23-s − 3.12e3·25-s + 3.71e3·26-s − 494.·28-s + 8.96e3·29-s + 5.85e3·31-s − 1.07e3·32-s + 3.97e3·34-s + 76.0·35-s − 55.9·37-s + 3.28e3·38-s + 86.0·40-s + 1.04e4·41-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.0927·4-s + 0.00816·5-s + 1.28·7-s + 1.04·8-s − 0.00778·10-s + 0.163·11-s − 1.13·13-s − 1.22·14-s − 0.898·16-s − 0.618·17-s − 0.387·19-s − 0.000757·20-s − 0.155·22-s − 0.517·23-s − 0.999·25-s + 1.07·26-s − 0.119·28-s + 1.98·29-s + 1.09·31-s − 0.184·32-s + 0.589·34-s + 0.0104·35-s − 0.00671·37-s + 0.369·38-s + 0.00850·40-s + 0.970·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 5.38T + 32T^{2} \) |
| 5 | \( 1 - 0.456T + 3.12e3T^{2} \) |
| 7 | \( 1 - 166.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 65.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 689.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 737.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 609.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 55.9T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.04e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.20e4T + 8.58e9T^{2} \) |
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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
−9.975177626208147828602750411702, −9.110118660519270731487522991846, −8.130342980155909315791606743057, −7.71457640488157044748234755802, −6.43178752514898571872612031596, −4.88868420867748512801971996448, −4.36836881655541215725590961009, −2.38565763911160218782649913177, −1.29792759077018804656899951417, 0,
1.29792759077018804656899951417, 2.38565763911160218782649913177, 4.36836881655541215725590961009, 4.88868420867748512801971996448, 6.43178752514898571872612031596, 7.71457640488157044748234755802, 8.130342980155909315791606743057, 9.110118660519270731487522991846, 9.975177626208147828602750411702
