| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 4·7-s + 8-s + 9-s − 10-s − 2·11-s − 12-s − 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s − 4·21-s − 2·22-s − 2·23-s − 24-s + 25-s − 2·26-s − 27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s − 0.426·22-s − 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.323133554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323133554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.54273301360745, −13.82534432676788, −13.51011239639429, −12.77403240032941, −12.26908378624378, −11.89399332605240, −11.38828399816747, −10.95501773608729, −10.61557026687739, −9.909644311897013, −9.264747946882898, −8.518807339782791, −7.971619042596217, −7.432846921103439, −7.227628896031534, −6.319890318037633, −5.696351701453492, −5.194138201400292, −4.785314114236317, −4.252438629344585, −3.688821429077725, −2.708353679847757, −2.208259337569423, −1.393231764422472, −0.5973971424706004,
0.5973971424706004, 1.393231764422472, 2.208259337569423, 2.708353679847757, 3.688821429077725, 4.252438629344585, 4.785314114236317, 5.194138201400292, 5.696351701453492, 6.319890318037633, 7.227628896031534, 7.432846921103439, 7.971619042596217, 8.518807339782791, 9.264747946882898, 9.909644311897013, 10.61557026687739, 10.95501773608729, 11.38828399816747, 11.89399332605240, 12.26908378624378, 12.77403240032941, 13.51011239639429, 13.82534432676788, 14.54273301360745
