| L(s) = 1 | − 0.723·2-s + 3·3-s − 7.47·4-s − 2.17·6-s + 1.13·7-s + 11.1·8-s + 9·9-s + 11·11-s − 22.4·12-s − 21.8·13-s − 0.819·14-s + 51.7·16-s + 6.18·17-s − 6.51·18-s − 92.8·19-s + 3.39·21-s − 7.96·22-s − 36.7·23-s + 33.5·24-s + 15.8·26-s + 27·27-s − 8.46·28-s + 71.1·29-s + 186.·31-s − 127.·32-s + 33·33-s − 4.47·34-s + ⋯ |
| L(s) = 1 | − 0.255·2-s + 0.577·3-s − 0.934·4-s − 0.147·6-s + 0.0611·7-s + 0.494·8-s + 0.333·9-s + 0.301·11-s − 0.539·12-s − 0.466·13-s − 0.0156·14-s + 0.807·16-s + 0.0881·17-s − 0.0852·18-s − 1.12·19-s + 0.0353·21-s − 0.0771·22-s − 0.333·23-s + 0.285·24-s + 0.119·26-s + 0.192·27-s − 0.0571·28-s + 0.455·29-s + 1.08·31-s − 0.701·32-s + 0.174·33-s − 0.0225·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 0.723T + 8T^{2} \) |
| 7 | \( 1 - 1.13T + 343T^{2} \) |
| 13 | \( 1 + 21.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.18T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 155.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 234.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 441.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 761.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 51.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 703.T + 9.12e5T^{2} \) |
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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
−9.303096338694315331886966347972, −8.616664110169027445670224872884, −7.962484579762390112228576164610, −6.99754085167477805901229589265, −5.85866678728864268153075339313, −4.66974320855044842980915030680, −4.03340274172898315513678322146, −2.78583340557904427382802754752, −1.43239054369412809949472506116, 0,
1.43239054369412809949472506116, 2.78583340557904427382802754752, 4.03340274172898315513678322146, 4.66974320855044842980915030680, 5.85866678728864268153075339313, 6.99754085167477805901229589265, 7.962484579762390112228576164610, 8.616664110169027445670224872884, 9.303096338694315331886966347972
