| L(s) = 1 | + 5-s + 7-s + 6·13-s − 2·17-s + 8·19-s − 8·23-s + 25-s − 2·29-s + 4·31-s + 35-s + 2·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s + 10·53-s + 4·59-s + 2·61-s + 6·65-s − 4·67-s + 12·71-s − 2·73-s + 8·79-s − 4·83-s − 2·85-s + 6·89-s + 6·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 0.520·59-s + 0.256·61-s + 0.744·65-s − 0.488·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105086988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105086988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.80186539217194, −15.13699850060373, −14.46636358127087, −13.88921963394307, −13.53958856390691, −13.17011881187244, −12.24979377854129, −11.77527590609099, −11.23715218750731, −10.77462101184671, −9.972869476106015, −9.622201777088913, −8.926235151357052, −8.227664804106067, −7.936911408730500, −7.048687495916990, −6.435780754273863, −5.780817701238060, −5.404578855409826, −4.487674336443913, −3.836802749502943, −3.205662830185003, −2.277712660698307, −1.518290287215120, −0.7756310195661762,
0.7756310195661762, 1.518290287215120, 2.277712660698307, 3.205662830185003, 3.836802749502943, 4.487674336443913, 5.404578855409826, 5.780817701238060, 6.435780754273863, 7.048687495916990, 7.936911408730500, 8.227664804106067, 8.926235151357052, 9.622201777088913, 9.972869476106015, 10.77462101184671, 11.23715218750731, 11.77527590609099, 12.24979377854129, 13.17011881187244, 13.53958856390691, 13.88921963394307, 14.46636358127087, 15.13699850060373, 15.80186539217194
