| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 27-s − 2·29-s + 8·31-s + 4·33-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s − 2·51-s + 10·53-s − 4·57-s − 12·59-s − 6·61-s + 12·67-s − 6·73-s + 8·79-s + 81-s − 4·83-s + 2·87-s − 2·89-s − 8·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.768·61-s + 1.46·67-s − 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s − 0.211·89-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.439422377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439422377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.24606905751142, −13.70522007723011, −13.42386733444841, −12.69112420401850, −12.27075771823837, −11.90746284374274, −11.23790778218500, −10.80142666715803, −10.22082975979885, −9.821417426940685, −9.373830809401450, −8.559938161386276, −7.975826797662010, −7.560615517965568, −7.062674648522052, −6.390442399044493, −5.687269963651377, −5.383068535834802, −4.742769202976670, −4.251147338656392, −3.323927891592937, −2.801708559383688, −2.136471367556430, −1.190552326090878, −0.4634921744827620,
0.4634921744827620, 1.190552326090878, 2.136471367556430, 2.801708559383688, 3.323927891592937, 4.251147338656392, 4.742769202976670, 5.383068535834802, 5.687269963651377, 6.390442399044493, 7.062674648522052, 7.560615517965568, 7.975826797662010, 8.559938161386276, 9.373830809401450, 9.821417426940685, 10.22082975979885, 10.80142666715803, 11.23790778218500, 11.90746284374274, 12.27075771823837, 12.69112420401850, 13.42386733444841, 13.70522007723011, 14.24606905751142
