L(s) = 1 | − 0.514·2-s − 1.73·4-s + 5-s − 4.44·7-s + 1.92·8-s − 0.514·10-s + 5.37·11-s + 13-s + 2.29·14-s + 2.47·16-s − 4.15·17-s − 2.50·19-s − 1.73·20-s − 2.76·22-s − 2.60·23-s + 25-s − 0.514·26-s + 7.71·28-s + 3.18·29-s + 4.19·31-s − 5.12·32-s + 2.14·34-s − 4.44·35-s − 5.57·37-s + 1.28·38-s + 1.92·40-s + 4.43·41-s + ⋯ |
L(s) = 1 | − 0.364·2-s − 0.867·4-s + 0.447·5-s − 1.68·7-s + 0.679·8-s − 0.162·10-s + 1.61·11-s + 0.277·13-s + 0.612·14-s + 0.619·16-s − 1.00·17-s − 0.574·19-s − 0.387·20-s − 0.589·22-s − 0.543·23-s + 0.200·25-s − 0.100·26-s + 1.45·28-s + 0.591·29-s + 0.753·31-s − 0.905·32-s + 0.367·34-s − 0.752·35-s − 0.916·37-s + 0.209·38-s + 0.304·40-s + 0.692·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.514T + 2T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 + 3.83T + 67T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 - 7.79T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−9.132398663109629576093011316586, −8.467115746821913354284347379435, −7.22776058415693999179402370956, −6.33694257305268691107416062898, −6.04541552605901642387485999866, −4.54191420520330268020915833243, −3.94371082930717565779829315528, −2.92475006805779607765706445735, −1.41996513375349376407094248914, 0,
1.41996513375349376407094248914, 2.92475006805779607765706445735, 3.94371082930717565779829315528, 4.54191420520330268020915833243, 6.04541552605901642387485999866, 6.33694257305268691107416062898, 7.22776058415693999179402370956, 8.467115746821913354284347379435, 9.132398663109629576093011316586