| L(s) = 1 | + 4·2-s − 27·3-s + 16·4-s − 108·6-s − 49·7-s + 64·8-s + 486·9-s − 525·11-s − 432·12-s + 551·13-s − 196·14-s + 256·16-s + 1.50e3·17-s + 1.94e3·18-s − 848·19-s + 1.32e3·21-s − 2.10e3·22-s + 2.39e3·23-s − 1.72e3·24-s + 2.20e3·26-s − 6.56e3·27-s − 784·28-s + 2.23e3·29-s − 7.27e3·31-s + 1.02e3·32-s + 1.41e4·33-s + 6.00e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.30·11-s − 0.866·12-s + 0.904·13-s − 0.267·14-s + 1/4·16-s + 1.25·17-s + 1.41·18-s − 0.538·19-s + 0.654·21-s − 0.925·22-s + 0.942·23-s − 0.612·24-s + 0.639·26-s − 1.73·27-s − 0.188·28-s + 0.494·29-s − 1.35·31-s + 0.176·32-s + 2.26·33-s + 0.890·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + p^{3} T + p^{5} T^{2} \) |
| 11 | \( 1 + 525 T + p^{5} T^{2} \) |
| 13 | \( 1 - 551 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1501 T + p^{5} T^{2} \) |
| 19 | \( 1 + 848 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2390 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2239 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7274 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12302 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9054 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13228 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20947 T + p^{5} T^{2} \) |
| 53 | \( 1 + 35334 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11974 T + p^{5} T^{2} \) |
| 61 | \( 1 + 20952 T + p^{5} T^{2} \) |
| 67 | \( 1 + 54614 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14160 T + p^{5} T^{2} \) |
| 73 | \( 1 + 4598 T + p^{5} T^{2} \) |
| 79 | \( 1 + 36727 T + p^{5} T^{2} \) |
| 83 | \( 1 + 84156 T + p^{5} T^{2} \) |
| 89 | \( 1 - 59584 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119595 T + p^{5} T^{2} \) |
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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
−10.70524371009618687432385931286, −9.678876381866035717771790109541, −7.985605545547717815730361369157, −6.94474021346310146456994865388, −5.98644297670880416635896599778, −5.42318053967170906108512140097, −4.47139034529521406689058964603, −3.10572552523826378627938623841, −1.29746693537960837273554653475, 0,
1.29746693537960837273554653475, 3.10572552523826378627938623841, 4.47139034529521406689058964603, 5.42318053967170906108512140097, 5.98644297670880416635896599778, 6.94474021346310146456994865388, 7.985605545547717815730361369157, 9.678876381866035717771790109541, 10.70524371009618687432385931286
