| L(s) = 1 | − 3·3-s − 4·7-s + 6·9-s − 3·11-s − 6·13-s + 2·17-s − 19-s + 12·21-s − 9·27-s − 2·29-s − 10·31-s + 9·33-s + 8·37-s + 18·39-s + 5·41-s + 11·43-s − 2·47-s + 9·49-s − 6·51-s − 12·53-s + 3·57-s − 13·59-s + 8·61-s − 24·63-s − 3·67-s − 10·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.51·7-s + 2·9-s − 0.904·11-s − 1.66·13-s + 0.485·17-s − 0.229·19-s + 2.61·21-s − 1.73·27-s − 0.371·29-s − 1.79·31-s + 1.56·33-s + 1.31·37-s + 2.88·39-s + 0.780·41-s + 1.67·43-s − 0.291·47-s + 9/7·49-s − 0.840·51-s − 1.64·53-s + 0.397·57-s − 1.69·59-s + 1.02·61-s − 3.02·63-s − 0.366·67-s − 1.18·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1004079529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1004079529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−12.72861715133725, −12.59995805534505, −12.26443512622450, −11.60932175150532, −11.02378175397329, −10.80630302953824, −10.20520128456753, −9.860586437033050, −9.346973409428103, −9.182479454741675, −7.984412235865017, −7.537776199675464, −7.243582472438104, −6.727104881180615, −6.019074743515096, −5.903068986761762, −5.354754240209491, −4.798158376035206, −4.372778740862705, −3.696607882889736, −2.964692687642750, −2.511886699071377, −1.732499795128772, −0.7898606431148208, −0.1345056780678856,
0.1345056780678856, 0.7898606431148208, 1.732499795128772, 2.511886699071377, 2.964692687642750, 3.696607882889736, 4.372778740862705, 4.798158376035206, 5.354754240209491, 5.903068986761762, 6.019074743515096, 6.727104881180615, 7.243582472438104, 7.537776199675464, 7.984412235865017, 9.182479454741675, 9.346973409428103, 9.860586437033050, 10.20520128456753, 10.80630302953824, 11.02378175397329, 11.60932175150532, 12.26443512622450, 12.59995805534505, 12.72861715133725
