| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s − 4·13-s + 14-s + 16-s − 18-s + 8·19-s + 21-s − 3·22-s + 3·23-s + 24-s + 4·26-s − 27-s − 28-s + 6·29-s − 2·31-s − 32-s − 3·33-s + 36-s − 5·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.218·21-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s − 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.915990360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915990360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.49722847112193, −12.05635654067409, −11.69766334682171, −11.48300029322697, −10.77727013876023, −10.27239532692222, −9.791417235413809, −9.727627100278310, −8.898091237576555, −8.787284828405798, −7.975133671199312, −7.483014880813965, −7.051829001086682, −6.704477428034431, −6.300513315544436, −5.540835894924870, −5.025883816987942, −4.915072613417977, −3.786603683393269, −3.579047617343809, −2.865350470213738, −2.254370693973345, −1.622412399640873, −0.8289410419056247, −0.5881895450229414,
0.5881895450229414, 0.8289410419056247, 1.622412399640873, 2.254370693973345, 2.865350470213738, 3.579047617343809, 3.786603683393269, 4.915072613417977, 5.025883816987942, 5.540835894924870, 6.300513315544436, 6.704477428034431, 7.051829001086682, 7.483014880813965, 7.975133671199312, 8.787284828405798, 8.898091237576555, 9.727627100278310, 9.791417235413809, 10.27239532692222, 10.77727013876023, 11.48300029322697, 11.69766334682171, 12.05635654067409, 12.49722847112193
