| L(s) = 1 | + 2·2-s − 28·4-s + 76·5-s + 49·7-s − 120·8-s + 152·10-s − 121·11-s − 402·13-s + 98·14-s + 656·16-s − 376·17-s + 2.03e3·19-s − 2.12e3·20-s − 242·22-s − 2.10e3·23-s + 2.65e3·25-s − 804·26-s − 1.37e3·28-s − 1.47e3·29-s − 3.87e3·31-s + 5.15e3·32-s − 752·34-s + 3.72e3·35-s − 2.86e3·37-s + 4.07e3·38-s − 9.12e3·40-s − 1.32e3·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s + 1.35·5-s + 0.377·7-s − 0.662·8-s + 0.480·10-s − 0.301·11-s − 0.659·13-s + 0.133·14-s + 0.640·16-s − 0.315·17-s + 1.29·19-s − 1.18·20-s − 0.106·22-s − 0.827·23-s + 0.848·25-s − 0.233·26-s − 0.330·28-s − 0.326·29-s − 0.724·31-s + 0.889·32-s − 0.111·34-s + 0.513·35-s − 0.343·37-s + 0.457·38-s − 0.901·40-s − 0.123·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 5 | \( 1 - 76 T + p^{5} T^{2} \) |
| 13 | \( 1 + 402 T + p^{5} T^{2} \) |
| 17 | \( 1 + 376 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2038 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2100 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1478 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3874 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2862 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1328 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8776 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3454 T + p^{5} T^{2} \) |
| 53 | \( 1 + 37134 T + p^{5} T^{2} \) |
| 59 | \( 1 - 37336 T + p^{5} T^{2} \) |
| 61 | \( 1 + 22426 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12852 T + p^{5} T^{2} \) |
| 71 | \( 1 - 33048 T + p^{5} T^{2} \) |
| 73 | \( 1 - 45848 T + p^{5} T^{2} \) |
| 79 | \( 1 + 5048 T + p^{5} T^{2} \) |
| 83 | \( 1 - 2794 T + p^{5} T^{2} \) |
| 89 | \( 1 - 46038 T + p^{5} T^{2} \) |
| 97 | \( 1 + 103558 T + p^{5} T^{2} \) |
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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
−9.607826962889712008631024458928, −8.521030005129515269213434274801, −7.56423329290688435245332178953, −6.34851337584256343523850780716, −5.39471286996938116068563782949, −4.99817638213722120732100595808, −3.71508665144201886550929924348, −2.50824799935553735647479518870, −1.42281568790477107379086027779, 0,
1.42281568790477107379086027779, 2.50824799935553735647479518870, 3.71508665144201886550929924348, 4.99817638213722120732100595808, 5.39471286996938116068563782949, 6.34851337584256343523850780716, 7.56423329290688435245332178953, 8.521030005129515269213434274801, 9.607826962889712008631024458928
