| L(s) = 1 | − 27·3-s − 722·7-s + 729·9-s − 3.99e3·11-s + 3.03e3·13-s − 2.05e4·17-s − 2.53e4·19-s + 1.94e4·21-s + 6.66e4·23-s − 1.96e4·27-s − 1.52e5·29-s − 1.23e5·31-s + 1.07e5·33-s − 3.37e5·37-s − 8.18e4·39-s + 3.96e5·41-s + 4.42e5·43-s + 1.70e5·47-s − 3.02e5·49-s + 5.55e5·51-s − 1.23e6·53-s + 6.83e5·57-s − 3.02e5·59-s − 2.83e6·61-s − 5.26e5·63-s + 3.74e6·67-s − 1.79e6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.795·7-s + 1/3·9-s − 0.904·11-s + 0.382·13-s − 1.01·17-s − 0.846·19-s + 0.459·21-s + 1.14·23-s − 0.192·27-s − 1.16·29-s − 0.746·31-s + 0.522·33-s − 1.09·37-s − 0.220·39-s + 0.898·41-s + 0.849·43-s + 0.239·47-s − 0.367·49-s + 0.586·51-s − 1.14·53-s + 0.488·57-s − 0.191·59-s − 1.59·61-s − 0.265·63-s + 1.51·67-s − 0.659·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7772893991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7772893991\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 722 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3994 T + p^{7} T^{2} \) |
| 13 | \( 1 - 3030 T + p^{7} T^{2} \) |
| 17 | \( 1 + 20582 T + p^{7} T^{2} \) |
| 19 | \( 1 + 25320 T + p^{7} T^{2} \) |
| 23 | \( 1 - 66652 T + p^{7} T^{2} \) |
| 29 | \( 1 + 152664 T + p^{7} T^{2} \) |
| 31 | \( 1 + 123776 T + p^{7} T^{2} \) |
| 37 | \( 1 + 337886 T + p^{7} T^{2} \) |
| 41 | \( 1 - 396530 T + p^{7} T^{2} \) |
| 43 | \( 1 - 442852 T + p^{7} T^{2} \) |
| 47 | \( 1 - 170432 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1239426 T + p^{7} T^{2} \) |
| 59 | \( 1 + 302354 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2830198 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3741272 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1007580 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2404636 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7517832 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5299628 T + p^{7} T^{2} \) |
| 89 | \( 1 + 7650250 T + p^{7} T^{2} \) |
| 97 | \( 1 + 10055944 T + p^{7} T^{2} \) |
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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
−10.81550329976118992790156329626, −9.600852117380396012879906076162, −8.754691432563297291190860314264, −7.47236876336626924993652887663, −6.56139379127429694032925567024, −5.65029340775368134779920307842, −4.55495229828159349016118049621, −3.31187597552597206658315868229, −2.01558871267874283564892514431, −0.42487923978744712926611776472,
0.42487923978744712926611776472, 2.01558871267874283564892514431, 3.31187597552597206658315868229, 4.55495229828159349016118049621, 5.65029340775368134779920307842, 6.56139379127429694032925567024, 7.47236876336626924993652887663, 8.754691432563297291190860314264, 9.600852117380396012879906076162, 10.81550329976118992790156329626
