| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 4·13-s + 14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s + 20-s − 21-s − 2·22-s − 24-s + 25-s − 4·26-s + 27-s − 28-s + 2·29-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 + 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.744953362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.744953362\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.52248319370096, −13.39110703533993, −12.66367045998871, −11.99640892019836, −11.83332933898336, −11.15908858535206, −10.45458016942295, −10.17364226241394, −9.611032998130027, −9.341211644504789, −8.653217912399284, −8.313191172413955, −7.846683859901760, −7.134207299396729, −6.713159518357817, −6.275746596493327, −5.429387147823411, −5.319091658313670, −4.180795778231830, −3.580893379101966, −3.261691347387741, −2.567465359802940, −1.808824358343346, −1.178336777522766, −0.7385062363901729,
0.7385062363901729, 1.178336777522766, 1.808824358343346, 2.567465359802940, 3.261691347387741, 3.580893379101966, 4.180795778231830, 5.319091658313670, 5.429387147823411, 6.275746596493327, 6.713159518357817, 7.134207299396729, 7.846683859901760, 8.313191172413955, 8.653217912399284, 9.341211644504789, 9.611032998130027, 10.17364226241394, 10.45458016942295, 11.15908858535206, 11.83332933898336, 11.99640892019836, 12.66367045998871, 13.39110703533993, 13.52248319370096
