| L(s) = 1 | + 2.73·2-s − 1.15·3-s + 5.47·4-s − 1.87·5-s − 3.16·6-s + 9.50·8-s − 1.66·9-s − 5.13·10-s − 2.29·11-s − 6.32·12-s + 2.17·15-s + 15.0·16-s − 6.07·17-s − 4.54·18-s + 5.15·19-s − 10.2·20-s − 6.28·22-s + 4.41·23-s − 10.9·24-s − 1.47·25-s + 5.39·27-s + 7.50·29-s + 5.93·30-s − 4.33·31-s + 22.0·32-s + 2.65·33-s − 16.6·34-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.667·3-s + 2.73·4-s − 0.839·5-s − 1.29·6-s + 3.36·8-s − 0.554·9-s − 1.62·10-s − 0.693·11-s − 1.82·12-s + 0.560·15-s + 3.75·16-s − 1.47·17-s − 1.07·18-s + 1.18·19-s − 2.29·20-s − 1.34·22-s + 0.921·23-s − 2.24·24-s − 0.295·25-s + 1.03·27-s + 1.39·29-s + 1.08·30-s − 0.778·31-s + 3.90·32-s + 0.462·33-s − 2.84·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.704535680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.704535680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 2.45T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 - 1.96T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 + 2.49T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.51T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42733578885640188588060162601, −6.86044285316242013415980447416, −6.27299959773210789998643939640, −5.38557451950651362465905797155, −5.14701344894435087439841340006, −4.34330033463085889619288265536, −3.70942275075333426010675056143, −2.85699393711179023158129435369, −2.30744205627352740408241566543, −0.799388967313065660980620821222,
0.799388967313065660980620821222, 2.30744205627352740408241566543, 2.85699393711179023158129435369, 3.70942275075333426010675056143, 4.34330033463085889619288265536, 5.14701344894435087439841340006, 5.38557451950651362465905797155, 6.27299959773210789998643939640, 6.86044285316242013415980447416, 7.42733578885640188588060162601
