| L(s) = 1 | + 2·5-s + 2·7-s + 6·11-s − 13-s + 2·17-s − 6·19-s − 25-s + 6·29-s + 6·31-s + 4·35-s + 2·37-s + 10·41-s + 8·43-s − 6·47-s − 3·49-s − 6·53-s + 12·55-s + 6·59-s − 10·61-s − 2·65-s + 2·67-s + 14·71-s − 14·73-s + 12·77-s + 4·79-s − 6·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.80·11-s − 0.277·13-s + 0.485·17-s − 1.37·19-s − 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.676·35-s + 0.328·37-s + 1.56·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s + 1.61·55-s + 0.781·59-s − 1.28·61-s − 0.248·65-s + 0.244·67-s + 1.66·71-s − 1.63·73-s + 1.36·77-s + 0.450·79-s − 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.874510826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.874510826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.487916109741254773551172049674, −7.935291927208856045100387693299, −6.82377369134986792965130796202, −6.31191344977864627789317544285, −5.66925387699141254194359268035, −4.54692527287818953154201346730, −4.14157227694140618939419170729, −2.83698851465638999201324281658, −1.87861151087067356201605101002, −1.08680531016401330966560066014,
1.08680531016401330966560066014, 1.87861151087067356201605101002, 2.83698851465638999201324281658, 4.14157227694140618939419170729, 4.54692527287818953154201346730, 5.66925387699141254194359268035, 6.31191344977864627789317544285, 6.82377369134986792965130796202, 7.935291927208856045100387693299, 8.487916109741254773551172049674
