| L(s) = 1 | − 3.23·3-s − 2·5-s − 7-s + 7.47·9-s + 11-s − 1.23·13-s + 6.47·15-s + 1.23·17-s + 2.47·19-s + 3.23·21-s + 6.47·23-s − 25-s − 14.4·27-s − 0.472·29-s + 7.23·31-s − 3.23·33-s + 2·35-s + 0.472·37-s + 4.00·39-s − 6.76·41-s − 8·43-s − 14.9·45-s − 7.23·47-s + 49-s − 4.00·51-s + 8.47·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s − 0.894·5-s − 0.377·7-s + 2.49·9-s + 0.301·11-s − 0.342·13-s + 1.67·15-s + 0.299·17-s + 0.567·19-s + 0.706·21-s + 1.34·23-s − 0.200·25-s − 2.78·27-s − 0.0876·29-s + 1.29·31-s − 0.563·33-s + 0.338·35-s + 0.0776·37-s + 0.640·39-s − 1.05·41-s − 1.21·43-s − 2.22·45-s − 1.05·47-s + 0.142·49-s − 0.560·51-s + 1.16·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.658007683079271225961522872242, −8.393942591645093172894639275399, −7.29469539437670736231617250406, −6.83102944525725550088189005455, −5.92833186216556798005243553184, −5.05889013210013028522599446080, −4.37470572462229169455552016936, −3.25944177071667590732404141922, −1.23675716743807956303806382658, 0,
1.23675716743807956303806382658, 3.25944177071667590732404141922, 4.37470572462229169455552016936, 5.05889013210013028522599446080, 5.92833186216556798005243553184, 6.83102944525725550088189005455, 7.29469539437670736231617250406, 8.393942591645093172894639275399, 9.658007683079271225961522872242
