| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 38·5-s + 36·6-s + 30·7-s + 64·8-s + 81·9-s − 152·10-s + 121·11-s + 144·12-s + 169·13-s + 120·14-s − 342·15-s + 256·16-s − 2.14e3·17-s + 324·18-s + 1.67e3·19-s − 608·20-s + 270·21-s + 484·22-s − 3.39e3·23-s + 576·24-s − 1.68e3·25-s + 676·26-s + 729·27-s + 480·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.679·5-s + 0.408·6-s + 0.231·7-s + 0.353·8-s + 1/3·9-s − 0.480·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.163·14-s − 0.392·15-s + 1/4·16-s − 1.80·17-s + 0.235·18-s + 1.06·19-s − 0.339·20-s + 0.133·21-s + 0.213·22-s − 1.33·23-s + 0.204·24-s − 0.537·25-s + 0.196·26-s + 0.192·27-s + 0.115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 38 T + p^{5} T^{2} \) |
| 7 | \( 1 - 30 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2148 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1674 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3392 T + p^{5} T^{2} \) |
| 29 | \( 1 + 576 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4304 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4966 T + p^{5} T^{2} \) |
| 41 | \( 1 - 13768 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12442 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18872 T + p^{5} T^{2} \) |
| 53 | \( 1 + 11346 T + p^{5} T^{2} \) |
| 59 | \( 1 - 40732 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1238 T + p^{5} T^{2} \) |
| 67 | \( 1 + 6028 T + p^{5} T^{2} \) |
| 71 | \( 1 + 22376 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21206 T + p^{5} T^{2} \) |
| 79 | \( 1 + 53258 T + p^{5} T^{2} \) |
| 83 | \( 1 + 49232 T + p^{5} T^{2} \) |
| 89 | \( 1 - 112410 T + p^{5} T^{2} \) |
| 97 | \( 1 + 178978 T + p^{5} T^{2} \) |
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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
−8.904493442065763352576954025906, −8.057078413585270331964952185537, −7.30726863955350149563130259080, −6.44133678261306628870935697814, −5.36265391860032920055502079098, −4.21932634460832073464642303087, −3.77032651470179976062557189542, −2.56168911309104605068813934781, −1.57188540438823378863332053973, 0,
1.57188540438823378863332053973, 2.56168911309104605068813934781, 3.77032651470179976062557189542, 4.21932634460832073464642303087, 5.36265391860032920055502079098, 6.44133678261306628870935697814, 7.30726863955350149563130259080, 8.057078413585270331964952185537, 8.904493442065763352576954025906
