| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 216·6-s + 988·7-s − 512·8-s + 729·9-s − 8.04e3·11-s + 1.72e3·12-s + 3.33e3·13-s − 7.90e3·14-s + 4.09e3·16-s − 6.58e3·17-s − 5.83e3·18-s − 2.74e4·19-s + 2.66e4·21-s + 6.43e4·22-s − 4.86e4·23-s − 1.38e4·24-s − 2.66e4·26-s + 1.96e4·27-s + 6.32e4·28-s − 1.32e5·29-s + 2.54e5·31-s − 3.27e4·32-s − 2.17e5·33-s + 5.26e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.08·7-s − 0.353·8-s + 1/3·9-s − 1.82·11-s + 0.288·12-s + 0.420·13-s − 0.769·14-s + 1/4·16-s − 0.324·17-s − 0.235·18-s − 0.917·19-s + 0.628·21-s + 1.28·22-s − 0.832·23-s − 0.204·24-s − 0.297·26-s + 0.192·27-s + 0.544·28-s − 1.00·29-s + 1.53·31-s − 0.176·32-s − 1.05·33-s + 0.229·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 988 T + p^{7} T^{2} \) |
| 11 | \( 1 + 8040 T + p^{7} T^{2} \) |
| 13 | \( 1 - 3334 T + p^{7} T^{2} \) |
| 17 | \( 1 + 6582 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4 p^{3} T + p^{7} T^{2} \) |
| 23 | \( 1 + 48600 T + p^{7} T^{2} \) |
| 29 | \( 1 + 4566 p T + p^{7} T^{2} \) |
| 31 | \( 1 - 254408 T + p^{7} T^{2} \) |
| 37 | \( 1 + 519434 T + p^{7} T^{2} \) |
| 41 | \( 1 - 92394 T + p^{7} T^{2} \) |
| 43 | \( 1 - 234532 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1277640 T + p^{7} T^{2} \) |
| 53 | \( 1 - 835278 T + p^{7} T^{2} \) |
| 59 | \( 1 + 3068760 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1009330 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3082172 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3666720 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1122866 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4128808 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4586556 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5763678 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6747554 T + p^{7} T^{2} \) |
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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
−10.81647120019923850289671773327, −10.31176750373379500873767773854, −8.850952364308724255776531108123, −8.134356231457044551792662399511, −7.38402599288737284555215843209, −5.77094648009166221108040892861, −4.42511936818038452768319705389, −2.70230978446782484537968963241, −1.68651854076483058439918690633, 0,
1.68651854076483058439918690633, 2.70230978446782484537968963241, 4.42511936818038452768319705389, 5.77094648009166221108040892861, 7.38402599288737284555215843209, 8.134356231457044551792662399511, 8.850952364308724255776531108123, 10.31176750373379500873767773854, 10.81647120019923850289671773327
