| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 14·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 28·10-s + 8·11-s + 12·12-s + 13·13-s − 14·14-s − 42·15-s + 16·16-s − 98·17-s + 18·18-s − 28·19-s − 56·20-s − 21·21-s + 16·22-s − 52·23-s + 24·24-s + 71·25-s + 26·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.25·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.885·10-s + 0.219·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.722·15-s + 1/4·16-s − 1.39·17-s + 0.235·18-s − 0.338·19-s − 0.626·20-s − 0.218·21-s + 0.155·22-s − 0.471·23-s + 0.204·24-s + 0.567·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 98 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 T + p^{3} T^{2} \) |
| 31 | \( 1 + 168 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 514 T + p^{3} T^{2} \) |
| 43 | \( 1 + 236 T + p^{3} T^{2} \) |
| 47 | \( 1 + 216 T + p^{3} T^{2} \) |
| 53 | \( 1 + 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 84 T + p^{3} T^{2} \) |
| 61 | \( 1 - 446 T + p^{3} T^{2} \) |
| 67 | \( 1 - 292 T + p^{3} T^{2} \) |
| 71 | \( 1 - 100 T + p^{3} T^{2} \) |
| 73 | \( 1 - 450 T + p^{3} T^{2} \) |
| 79 | \( 1 - 392 T + p^{3} T^{2} \) |
| 83 | \( 1 + 292 T + p^{3} T^{2} \) |
| 89 | \( 1 + 402 T + p^{3} T^{2} \) |
| 97 | \( 1 - 314 T + p^{3} T^{2} \) |
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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
−10.04832708693777362225964514045, −8.836912419593144552396643740322, −8.172120309282781827993522153610, −7.12853941555572331444243648293, −6.45270833458583182972171734095, −4.97699619305406130594602751256, −3.97123283754570195238772543311, −3.37450189950249083125368556419, −1.97638427099841290753278357310, 0,
1.97638427099841290753278357310, 3.37450189950249083125368556419, 3.97123283754570195238772543311, 4.97699619305406130594602751256, 6.45270833458583182972171734095, 7.12853941555572331444243648293, 8.172120309282781827993522153610, 8.836912419593144552396643740322, 10.04832708693777362225964514045
