| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 5.86e3·5-s + 3.98e3·7-s + 3.27e4·8-s − 1.87e5·10-s + 5.00e5·11-s − 5.38e5·13-s + 1.27e5·14-s + 1.04e6·16-s − 1.84e6·17-s − 1.04e7·19-s − 6.00e6·20-s + 1.60e7·22-s − 4.05e7·23-s − 1.44e7·25-s − 1.72e7·26-s + 4.07e6·28-s − 4.11e7·29-s + 5.78e7·31-s + 3.35e7·32-s − 5.91e7·34-s − 2.33e7·35-s − 6.88e8·37-s − 3.34e8·38-s − 1.92e8·40-s − 5.80e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.839·5-s + 0.0895·7-s + 0.353·8-s − 0.593·10-s + 0.936·11-s − 0.401·13-s + 0.0633·14-s + 1/4·16-s − 0.315·17-s − 0.969·19-s − 0.419·20-s + 0.662·22-s − 1.31·23-s − 0.295·25-s − 0.284·26-s + 0.0447·28-s − 0.372·29-s + 0.363·31-s + 0.176·32-s − 0.223·34-s − 0.0751·35-s − 1.63·37-s − 0.685·38-s − 0.296·40-s − 0.782·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1173 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 569 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 500433 T + p^{11} T^{2} \) |
| 13 | \( 1 + 538012 T + p^{11} T^{2} \) |
| 17 | \( 1 + 1847880 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10461022 T + p^{11} T^{2} \) |
| 23 | \( 1 + 40549182 T + p^{11} T^{2} \) |
| 29 | \( 1 + 41169954 T + p^{11} T^{2} \) |
| 31 | \( 1 - 57873005 T + p^{11} T^{2} \) |
| 37 | \( 1 + 688582366 T + p^{11} T^{2} \) |
| 41 | \( 1 + 580836894 T + p^{11} T^{2} \) |
| 43 | \( 1 + 73179490 T + p^{11} T^{2} \) |
| 47 | \( 1 - 929053878 T + p^{11} T^{2} \) |
| 53 | \( 1 - 2611072053 T + p^{11} T^{2} \) |
| 59 | \( 1 - 4666302732 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7930719832 T + p^{11} T^{2} \) |
| 67 | \( 1 + 18493811722 T + p^{11} T^{2} \) |
| 71 | \( 1 + 16075533240 T + p^{11} T^{2} \) |
| 73 | \( 1 + 25404121951 T + p^{11} T^{2} \) |
| 79 | \( 1 - 34794772952 T + p^{11} T^{2} \) |
| 83 | \( 1 - 24133917129 T + p^{11} T^{2} \) |
| 89 | \( 1 - 1666560942 T + p^{11} T^{2} \) |
| 97 | \( 1 - 82667879663 T + p^{11} T^{2} \) |
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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
−12.24052724189689894227919479652, −11.63266170882240825965650080772, −10.30418301944012076888013466537, −8.672718533453841967092523080198, −7.38254197272081090740372139438, −6.16360927304094879885389528431, −4.54355940050962085328008613489, −3.59554266922157399458816031548, −1.88694331256200083251862008244, 0,
1.88694331256200083251862008244, 3.59554266922157399458816031548, 4.54355940050962085328008613489, 6.16360927304094879885389528431, 7.38254197272081090740372139438, 8.672718533453841967092523080198, 10.30418301944012076888013466537, 11.63266170882240825965650080772, 12.24052724189689894227919479652
