| L(s) = 1 | + 0.563·3-s + 2.34·5-s − 1.69·7-s − 2.68·9-s − 11-s + 4.11·13-s + 1.32·15-s − 6.16·17-s + 19-s − 0.953·21-s − 3.52·23-s + 0.502·25-s − 3.20·27-s − 8.10·29-s + 2.30·31-s − 0.563·33-s − 3.96·35-s − 6.56·37-s + 2.31·39-s + 7.75·41-s − 7.75·43-s − 6.29·45-s + 10.8·47-s − 4.13·49-s − 3.47·51-s + 7.93·53-s − 2.34·55-s + ⋯ |
| L(s) = 1 | + 0.325·3-s + 1.04·5-s − 0.639·7-s − 0.894·9-s − 0.301·11-s + 1.14·13-s + 0.341·15-s − 1.49·17-s + 0.229·19-s − 0.208·21-s − 0.734·23-s + 0.100·25-s − 0.616·27-s − 1.50·29-s + 0.413·31-s − 0.0980·33-s − 0.671·35-s − 1.07·37-s + 0.370·39-s + 1.21·41-s − 1.18·43-s − 0.938·45-s + 1.58·47-s − 0.590·49-s − 0.486·51-s + 1.08·53-s − 0.316·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.563T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 6.56T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 + 9.37T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.462463696706380694237349524291, −7.53589728432289678797071089099, −6.56177144863667410245759025340, −5.95915583974698970409716259686, −5.50279502111068171439298385585, −4.26678450715024770039773239756, −3.38474960759006691610551112364, −2.50024688272955491319156358870, −1.71141064521603140284658458544, 0,
1.71141064521603140284658458544, 2.50024688272955491319156358870, 3.38474960759006691610551112364, 4.26678450715024770039773239756, 5.50279502111068171439298385585, 5.95915583974698970409716259686, 6.56177144863667410245759025340, 7.53589728432289678797071089099, 8.462463696706380694237349524291
