| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 7-s + 9-s − 2·10-s − 11-s + 2·12-s − 2·14-s − 15-s − 4·16-s − 7·17-s + 2·18-s + 5·19-s − 2·20-s − 21-s − 2·22-s − 23-s + 25-s + 27-s − 2·28-s − 5·29-s − 2·30-s + 8·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 0.534·14-s − 0.258·15-s − 16-s − 1.69·17-s + 0.471·18-s + 1.14·19-s − 0.447·20-s − 0.218·21-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.928·29-s − 0.365·30-s + 1.43·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839814737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839814737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.20982345935799, −12.85958912910649, −12.24538570629103, −11.70515873567999, −11.45515448080541, −11.04240701935747, −10.17829866174542, −9.842542598155040, −9.356732692934801, −8.576427730918877, −8.470564433604749, −7.768909087465021, −7.092201927271200, −6.719874738678093, −6.382757611952393, −5.633807683065518, −5.115264520358327, −4.651114217934606, −4.222791950775241, −3.648383979976396, −3.032664812221014, −2.910234240232478, −2.039889247152216, −1.501237808496739, −0.2648439830816888,
0.2648439830816888, 1.501237808496739, 2.039889247152216, 2.910234240232478, 3.032664812221014, 3.648383979976396, 4.222791950775241, 4.651114217934606, 5.115264520358327, 5.633807683065518, 6.382757611952393, 6.719874738678093, 7.092201927271200, 7.768909087465021, 8.470564433604749, 8.576427730918877, 9.356732692934801, 9.842542598155040, 10.17829866174542, 11.04240701935747, 11.45515448080541, 11.70515873567999, 12.24538570629103, 12.85958912910649, 13.20982345935799
