| L(s) = 1 | − 3·3-s + 6·9-s + 2·11-s − 4·17-s − 6·19-s − 3·23-s − 9·27-s − 9·29-s + 4·31-s − 6·33-s − 4·37-s + 7·41-s − 5·43-s + 8·47-s + 12·51-s − 2·53-s + 18·57-s + 10·59-s + 61-s − 9·67-s + 9·69-s + 2·71-s − 4·73-s + 10·79-s + 9·81-s + 7·83-s + 27·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 0.603·11-s − 0.970·17-s − 1.37·19-s − 0.625·23-s − 1.73·27-s − 1.67·29-s + 0.718·31-s − 1.04·33-s − 0.657·37-s + 1.09·41-s − 0.762·43-s + 1.16·47-s + 1.68·51-s − 0.274·53-s + 2.38·57-s + 1.30·59-s + 0.128·61-s − 1.09·67-s + 1.08·69-s + 0.237·71-s − 0.468·73-s + 1.12·79-s + 81-s + 0.768·83-s + 2.89·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4456699122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4456699122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.93036212693055, −13.27146573228183, −12.97817451275111, −12.38687235210266, −11.98984730381010, −11.48604134548891, −11.07129284843442, −10.67216318174498, −10.17754553112205, −9.656464918118505, −8.917845378290674, −8.618599082689553, −7.640930589607672, −7.277547367547160, −6.529888733858371, −6.280520147505862, −5.859001106056115, −5.116510858242835, −4.700318357638145, −3.988374474085488, −3.764573422862098, −2.441493767247084, −1.915684374387636, −1.124180017327280, −0.2689228444558571,
0.2689228444558571, 1.124180017327280, 1.915684374387636, 2.441493767247084, 3.764573422862098, 3.988374474085488, 4.700318357638145, 5.116510858242835, 5.859001106056115, 6.280520147505862, 6.529888733858371, 7.277547367547160, 7.640930589607672, 8.618599082689553, 8.917845378290674, 9.656464918118505, 10.17754553112205, 10.67216318174498, 11.07129284843442, 11.48604134548891, 11.98984730381010, 12.38687235210266, 12.97817451275111, 13.27146573228183, 13.93036212693055
