| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s + 6·13-s − 14-s − 15-s + 16-s + 5·17-s + 18-s + 8·19-s + 20-s + 21-s − 3·22-s − 23-s − 24-s + 25-s + 6·26-s − 27-s − 28-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 + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.152961539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.152961539\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.94340894049008, −12.80112016486870, −12.10563783247768, −11.62862104331585, −11.36655345580606, −10.72285422194340, −10.25130044523386, −10.02568371872553, −9.298031278343832, −8.951990759163478, −8.023468094365803, −7.764242223879287, −7.317543881901708, −6.572313032196244, −6.024408659794602, −5.865719788614586, −5.349122078680262, −4.849883756284595, −4.218683780836710, −3.576952894184065, −3.053953552442348, −2.751359004158096, −1.707196696496782, −1.194694600969045, −0.6440581441390026,
0.6440581441390026, 1.194694600969045, 1.707196696496782, 2.751359004158096, 3.053953552442348, 3.576952894184065, 4.218683780836710, 4.849883756284595, 5.349122078680262, 5.865719788614586, 6.024408659794602, 6.572313032196244, 7.317543881901708, 7.764242223879287, 8.023468094365803, 8.951990759163478, 9.298031278343832, 10.02568371872553, 10.25130044523386, 10.72285422194340, 11.36655345580606, 11.62862104331585, 12.10563783247768, 12.80112016486870, 12.94340894049008
