| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s + 8·8-s + 9·9-s − 4·10-s − 8·11-s − 12·12-s + 42·13-s + 6·15-s + 16·16-s + 2·17-s + 18·18-s + 124·19-s − 8·20-s − 16·22-s + 76·23-s − 24·24-s − 121·25-s + 84·26-s − 27·27-s + 254·29-s + 12·30-s + 72·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.178·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.126·10-s − 0.219·11-s − 0.288·12-s + 0.896·13-s + 0.103·15-s + 1/4·16-s + 0.0285·17-s + 0.235·18-s + 1.49·19-s − 0.0894·20-s − 0.155·22-s + 0.689·23-s − 0.204·24-s − 0.967·25-s + 0.633·26-s − 0.192·27-s + 1.62·29-s + 0.0730·30-s + 0.417·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.444877403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444877403\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 254 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 212 T + p^{3} T^{2} \) |
| 47 | \( 1 - 264 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 772 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 236 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1190 T + p^{3} T^{2} \) |
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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
−11.56886698311922542653532608853, −10.63851879848623807711697193557, −9.673140973884217385418283947736, −8.299091446871014868611931534960, −7.24487164749901783641199687735, −6.20217852812527652318977227994, −5.30256143482064034171982614073, −4.20281487194666214110630342027, −2.94703327487831006010805444320, −1.09334361875456715075271536774,
1.09334361875456715075271536774, 2.94703327487831006010805444320, 4.20281487194666214110630342027, 5.30256143482064034171982614073, 6.20217852812527652318977227994, 7.24487164749901783641199687735, 8.299091446871014868611931534960, 9.673140973884217385418283947736, 10.63851879848623807711697193557, 11.56886698311922542653532608853
