| L(s) = 1 | + 2.13·3-s − 5-s + 0.878·7-s + 1.55·9-s − 4.50·11-s − 2.05·13-s − 2.13·15-s − 4.31·17-s + 19-s + 1.87·21-s − 2.15·23-s + 25-s − 3.07·27-s + 9.01·29-s + 4.35·31-s − 9.62·33-s − 0.878·35-s − 8.67·37-s − 4.38·39-s + 4.26·41-s − 12.3·43-s − 1.55·45-s + 2.76·47-s − 6.22·49-s − 9.21·51-s − 3.98·53-s + 4.50·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.447·5-s + 0.331·7-s + 0.519·9-s − 1.35·11-s − 0.569·13-s − 0.551·15-s − 1.04·17-s + 0.229·19-s + 0.409·21-s − 0.449·23-s + 0.200·25-s − 0.592·27-s + 1.67·29-s + 0.781·31-s − 1.67·33-s − 0.148·35-s − 1.42·37-s − 0.702·39-s + 0.666·41-s − 1.88·43-s − 0.232·45-s + 0.403·47-s − 0.889·49-s − 1.29·51-s − 0.547·53-s + 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 7 | \( 1 - 0.878T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 + 3.98T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 7.55T + 97T^{2} \) |
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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
−8.404720077458391583508776466282, −7.76619960316089096326216491120, −7.14014929515008368467653506932, −6.14452903604937449207852780454, −4.94918347073555387524553622767, −4.49406180149230968746013697756, −3.25763914098061505575982546630, −2.73140779158461928123933634897, −1.81971563627221143955425862428, 0,
1.81971563627221143955425862428, 2.73140779158461928123933634897, 3.25763914098061505575982546630, 4.49406180149230968746013697756, 4.94918347073555387524553622767, 6.14452903604937449207852780454, 7.14014929515008368467653506932, 7.76619960316089096326216491120, 8.404720077458391583508776466282
