| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s − 2·9-s − 10-s + 12-s + 5·13-s − 14-s + 15-s + 16-s + 2·18-s + 5·19-s + 20-s + 21-s + 9·23-s − 24-s + 25-s − 5·26-s − 5·27-s + 28-s − 6·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s + 1.87·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.407659174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407659174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.978062143444101358885744521321, −7.26806416985595198145495430196, −6.49256694829097506387297488134, −5.77731833118557149105886673635, −5.20641496397529709615913146333, −4.12205827907315872846461654017, −3.13325972512981293431358301253, −2.73590226829810721728652383455, −1.59042127408644996677510922764, −0.886147682868866137306656548516,
0.886147682868866137306656548516, 1.59042127408644996677510922764, 2.73590226829810721728652383455, 3.13325972512981293431358301253, 4.12205827907315872846461654017, 5.20641496397529709615913146333, 5.77731833118557149105886673635, 6.49256694829097506387297488134, 7.26806416985595198145495430196, 7.978062143444101358885744521321
