| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 4·5-s + 36·6-s − 8·7-s − 64·8-s + 81·9-s − 16·10-s + 470·11-s − 144·12-s − 169·13-s + 32·14-s − 36·15-s + 256·16-s − 270·17-s − 324·18-s − 2.68e3·19-s + 64·20-s + 72·21-s − 1.88e3·22-s − 2.63e3·23-s + 576·24-s − 3.10e3·25-s + 676·26-s − 729·27-s − 128·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.0715·5-s + 0.408·6-s − 0.0617·7-s − 0.353·8-s + 1/3·9-s − 0.0505·10-s + 1.17·11-s − 0.288·12-s − 0.277·13-s + 0.0436·14-s − 0.0413·15-s + 1/4·16-s − 0.226·17-s − 0.235·18-s − 1.70·19-s + 0.0357·20-s + 0.0356·21-s − 0.828·22-s − 1.03·23-s + 0.204·24-s − 0.994·25-s + 0.196·26-s − 0.192·27-s − 0.0308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 4 T + p^{5} T^{2} \) |
| 7 | \( 1 + 8 T + p^{5} T^{2} \) |
| 11 | \( 1 - 470 T + p^{5} T^{2} \) |
| 17 | \( 1 + 270 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2688 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2636 T + p^{5} T^{2} \) |
| 29 | \( 1 + 950 T + p^{5} T^{2} \) |
| 31 | \( 1 + 284 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15978 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6324 T + p^{5} T^{2} \) |
| 43 | \( 1 - 6916 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5810 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1986 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7210 T + p^{5} T^{2} \) |
| 61 | \( 1 - 17050 T + p^{5} T^{2} \) |
| 67 | \( 1 + 28652 T + p^{5} T^{2} \) |
| 71 | \( 1 - 40970 T + p^{5} T^{2} \) |
| 73 | \( 1 + 56002 T + p^{5} T^{2} \) |
| 79 | \( 1 - 16328 T + p^{5} T^{2} \) |
| 83 | \( 1 + 49962 T + p^{5} T^{2} \) |
| 89 | \( 1 + 62304 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119722 T + p^{5} T^{2} \) |
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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
−12.60772043250875694950312104873, −11.70715460487360317747577025507, −10.64699266074999535464824415217, −9.585988899845665411616675875103, −8.428094522865072623543017545078, −6.95850646235105632652177115516, −5.95844252411820372949012293773, −4.09773098603447687162852988534, −1.84161534534573102855563261546, 0,
1.84161534534573102855563261546, 4.09773098603447687162852988534, 5.95844252411820372949012293773, 6.95850646235105632652177115516, 8.428094522865072623543017545078, 9.585988899845665411616675875103, 10.64699266074999535464824415217, 11.70715460487360317747577025507, 12.60772043250875694950312104873
