| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 6·11-s − 4·13-s − 19-s − 4·21-s + 3·23-s + 4·27-s + 8·31-s − 12·33-s + 37-s + 8·39-s + 3·41-s − 43-s − 6·47-s − 3·49-s − 9·53-s + 2·57-s − 3·59-s + 14·61-s + 2·63-s − 4·67-s − 6·69-s − 6·71-s + 11·73-s + 12·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.229·19-s − 0.872·21-s + 0.625·23-s + 0.769·27-s + 1.43·31-s − 2.08·33-s + 0.164·37-s + 1.28·39-s + 0.468·41-s − 0.152·43-s − 0.875·47-s − 3/7·49-s − 1.23·53-s + 0.264·57-s − 0.390·59-s + 1.79·61-s + 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.712·71-s + 1.28·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367872819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367872819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.506863405659471229977930285776, −7.74407899803003989638143010044, −6.67804883401462601781504454328, −6.50108919407440481405845477859, −5.46337353224654334009786014845, −4.77402321633752819724827727350, −4.22545534036621367303405162766, −2.99256630433841580776509883235, −1.73344867858359236084414018866, −0.75607631634855246623818726703,
0.75607631634855246623818726703, 1.73344867858359236084414018866, 2.99256630433841580776509883235, 4.22545534036621367303405162766, 4.77402321633752819724827727350, 5.46337353224654334009786014845, 6.50108919407440481405845477859, 6.67804883401462601781504454328, 7.74407899803003989638143010044, 8.506863405659471229977930285776
