| L(s) = 1 | + 1.85·3-s + 3.00·5-s + 0.497·7-s + 0.434·9-s + 5.10·11-s + 1.20·13-s + 5.56·15-s + 1.51·17-s − 4.66·19-s + 0.922·21-s − 1.17·23-s + 4.01·25-s − 4.75·27-s − 8.43·29-s + 4.49·31-s + 9.46·33-s + 1.49·35-s − 0.519·37-s + 2.23·39-s − 6.36·41-s + 11.7·43-s + 1.30·45-s − 7.73·47-s − 6.75·49-s + 2.80·51-s + 8.72·53-s + 15.3·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s + 1.34·5-s + 0.188·7-s + 0.144·9-s + 1.54·11-s + 0.334·13-s + 1.43·15-s + 0.366·17-s − 1.07·19-s + 0.201·21-s − 0.244·23-s + 0.802·25-s − 0.915·27-s − 1.56·29-s + 0.806·31-s + 1.64·33-s + 0.252·35-s − 0.0854·37-s + 0.357·39-s − 0.994·41-s + 1.79·43-s + 0.194·45-s − 1.12·47-s − 0.964·49-s + 0.392·51-s + 1.19·53-s + 2.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.201655846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.201655846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 3.00T + 5T^{2} \) |
| 7 | \( 1 - 0.497T + 7T^{2} \) |
| 11 | \( 1 - 5.10T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 8.43T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 0.519T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 - 8.02T + 59T^{2} \) |
| 61 | \( 1 + 0.243T + 61T^{2} \) |
| 67 | \( 1 + 0.980T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.526T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 2.26T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−9.489587013376555194136992633470, −8.875417012181042029660948618837, −8.294093768318186947722274347973, −7.16574262477754416183893085159, −6.24707547193024194807119285393, −5.65380092009169409005831420396, −4.29807884779379668126272686577, −3.42938689909358302583377837817, −2.26067525107499966709948261831, −1.52816439262473537684503051667,
1.52816439262473537684503051667, 2.26067525107499966709948261831, 3.42938689909358302583377837817, 4.29807884779379668126272686577, 5.65380092009169409005831420396, 6.24707547193024194807119285393, 7.16574262477754416183893085159, 8.294093768318186947722274347973, 8.875417012181042029660948618837, 9.489587013376555194136992633470
