| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 216·6-s + 1.12e3·7-s − 512·8-s + 729·9-s − 5.51e3·11-s − 1.72e3·12-s − 1.27e4·13-s − 9.00e3·14-s + 4.09e3·16-s + 3.22e4·17-s − 5.83e3·18-s − 4.44e3·19-s − 3.04e4·21-s + 4.41e4·22-s + 9.54e4·23-s + 1.38e4·24-s + 1.02e5·26-s − 1.96e4·27-s + 7.20e4·28-s + 1.94e4·29-s − 2.40e5·31-s − 3.27e4·32-s + 1.48e5·33-s − 2.57e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.24·7-s − 0.353·8-s + 1/3·9-s − 1.24·11-s − 0.288·12-s − 1.61·13-s − 0.877·14-s + 1/4·16-s + 1.58·17-s − 0.235·18-s − 0.148·19-s − 0.716·21-s + 0.883·22-s + 1.63·23-s + 0.204·24-s + 1.14·26-s − 0.192·27-s + 0.620·28-s + 0.148·29-s − 1.44·31-s − 0.176·32-s + 0.721·33-s − 1.12·34-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 | \( 1 - 1126 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5518 T + p^{7} T^{2} \) |
| 13 | \( 1 + 12798 T + p^{7} T^{2} \) |
| 17 | \( 1 - 32206 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4440 T + p^{7} T^{2} \) |
| 23 | \( 1 - 95452 T + p^{7} T^{2} \) |
| 29 | \( 1 - 19440 T + p^{7} T^{2} \) |
| 31 | \( 1 + 240248 T + p^{7} T^{2} \) |
| 37 | \( 1 + 77834 T + p^{7} T^{2} \) |
| 41 | \( 1 - 299522 T + p^{7} T^{2} \) |
| 43 | \( 1 - 416212 T + p^{7} T^{2} \) |
| 47 | \( 1 - 322976 T + p^{7} T^{2} \) |
| 53 | \( 1 + 880878 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1845110 T + p^{7} T^{2} \) |
| 61 | \( 1 + 861718 T + p^{7} T^{2} \) |
| 67 | \( 1 + 673864 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3426948 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4678748 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3137760 T + p^{7} T^{2} \) |
| 83 | \( 1 - 484132 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6258710 T + p^{7} T^{2} \) |
| 97 | \( 1 - 8657576 T + p^{7} T^{2} \) |
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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
−10.98011576773272423000279665539, −10.32988064182692716188077073000, −9.186434659047989970153627051697, −7.74464548330772264926226270744, −7.38815030634840109665379961838, −5.54882663271926651205320758277, −4.84046004725109938798951389987, −2.74636154900655326346604413238, −1.36617533883969395014554057589, 0,
1.36617533883969395014554057589, 2.74636154900655326346604413238, 4.84046004725109938798951389987, 5.54882663271926651205320758277, 7.38815030634840109665379961838, 7.74464548330772264926226270744, 9.186434659047989970153627051697, 10.32988064182692716188077073000, 10.98011576773272423000279665539
