| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 4·14-s − 15-s + 16-s + 4·17-s − 18-s − 4·19-s + 20-s − 4·21-s − 4·22-s + 8·23-s + 24-s + 25-s − 27-s + 4·28-s + 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.543571964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543571964\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.76823553350873, −12.41197956425972, −11.82371937684354, −11.54893779593862, −11.15173442532591, −10.63733686549137, −10.19385394625086, −9.824998682623040, −9.115790966912702, −8.781302617742807, −8.193426599545219, −8.015252023442904, −7.139985049822716, −6.748934057339226, −6.479878613857230, −5.733634665795967, −5.218473319827813, −4.778951903376837, −4.357449205335573, −3.498077999050537, −3.017045134942771, −1.956369999100424, −1.834788301658014, −0.9178441433708898, −0.8194970709009295,
0.8194970709009295, 0.9178441433708898, 1.834788301658014, 1.956369999100424, 3.017045134942771, 3.498077999050537, 4.357449205335573, 4.778951903376837, 5.218473319827813, 5.733634665795967, 6.479878613857230, 6.748934057339226, 7.139985049822716, 8.015252023442904, 8.193426599545219, 8.781302617742807, 9.115790966912702, 9.824998682623040, 10.19385394625086, 10.63733686549137, 11.15173442532591, 11.54893779593862, 11.82371937684354, 12.41197956425972, 12.76823553350873
