| L(s) = 1 | + 3·3-s − 5·5-s + 9·9-s − 2·11-s + 14·13-s − 15·15-s + 26·17-s − 14·23-s + 25·25-s + 27·27-s + 38·29-s + 58·31-s − 6·33-s − 34·37-s + 42·39-s − 74·43-s − 45·45-s + 34·47-s + 49·49-s + 78·51-s + 10·55-s − 98·59-s − 70·65-s − 26·67-s − 42·69-s + 75·75-s − 38·79-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 9-s − 0.181·11-s + 1.07·13-s − 15-s + 1.52·17-s − 0.608·23-s + 25-s + 27-s + 1.31·29-s + 1.87·31-s − 0.181·33-s − 0.918·37-s + 1.07·39-s − 1.72·43-s − 45-s + 0.723·47-s + 49-s + 1.52·51-s + 2/11·55-s − 1.66·59-s − 1.07·65-s − 0.388·67-s − 0.608·69-s + 75-s − 0.481·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.295905968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295905968\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 2 T + p^{2} T^{2} \) |
| 13 | \( 1 - 14 T + p^{2} T^{2} \) |
| 17 | \( 1 - 26 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 14 T + p^{2} T^{2} \) |
| 29 | \( 1 - 38 T + p^{2} T^{2} \) |
| 31 | \( 1 - 58 T + p^{2} T^{2} \) |
| 37 | \( 1 + 34 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 74 T + p^{2} T^{2} \) |
| 47 | \( 1 - 34 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( 1 + 98 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 26 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 38 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−10.58187397486227512261130966319, −9.957102428943649857408336699973, −8.656477037431087848859219155829, −8.208146204477692914817212671585, −7.39829897577215458151536258345, −6.29514927400575730529465741414, −4.78031254155152240190851649084, −3.73088082099417128555400558610, −2.94455336575366994294748128384, −1.16795928830741876057529427343,
1.16795928830741876057529427343, 2.94455336575366994294748128384, 3.73088082099417128555400558610, 4.78031254155152240190851649084, 6.29514927400575730529465741414, 7.39829897577215458151536258345, 8.208146204477692914817212671585, 8.656477037431087848859219155829, 9.957102428943649857408336699973, 10.58187397486227512261130966319
