| L(s) = 1 | + 2·3-s + 1.73·5-s + 3.46·7-s + 9-s − 3.46·11-s − 5·13-s + 3.46·15-s − 6.92·17-s − 3.46·19-s + 6.92·21-s − 2.00·25-s − 4·27-s − 3·29-s + 8·31-s − 6.92·33-s + 5.99·35-s − 10·39-s − 9·41-s − 6.92·43-s + 1.73·45-s + 6·47-s + 4.99·49-s − 13.8·51-s − 1.73·53-s − 5.99·55-s − 6.92·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.774·5-s + 1.30·7-s + 0.333·9-s − 1.04·11-s − 1.38·13-s + 0.894·15-s − 1.68·17-s − 0.794·19-s + 1.51·21-s − 0.400·25-s − 0.769·27-s − 0.557·29-s + 1.43·31-s − 1.20·33-s + 1.01·35-s − 1.60·39-s − 1.40·41-s − 1.05·43-s + 0.258·45-s + 0.875·47-s + 0.714·49-s − 1.94·51-s − 0.237·53-s − 0.809·55-s − 0.917·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.62895164715173360183761853026, −6.93414360914174670556141217730, −6.06760826370947008502471329901, −5.08141442666137673795662028681, −4.79347122975735307494928663586, −3.89131910924881802239585761496, −2.66949487106070280373090640400, −2.28551586973661676352614618645, −1.77380623000668477939532424909, 0,
1.77380623000668477939532424909, 2.28551586973661676352614618645, 2.66949487106070280373090640400, 3.89131910924881802239585761496, 4.79347122975735307494928663586, 5.08141442666137673795662028681, 6.06760826370947008502471329901, 6.93414360914174670556141217730, 7.62895164715173360183761853026
