| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 19-s + 20-s + 21-s − 4·22-s + 8·23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.347933207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347933207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.55494843360745, −14.03201879909776, −13.26568208360352, −12.95627017034955, −12.38269176471765, −11.72492899113354, −11.42184909359489, −10.79713849579322, −10.41356381564899, −9.671813855980365, −9.366251916891545, −8.857583705300642, −8.409294639882072, −7.378806384099444, −7.105595062893111, −6.562586641968240, −6.103083756274024, −5.407246019539930, −4.905445897440579, −4.024457167545261, −3.517429639313625, −2.640088868178759, −1.943027965169244, −1.243356579677029, −0.5097769304333557,
0.5097769304333557, 1.243356579677029, 1.943027965169244, 2.640088868178759, 3.517429639313625, 4.024457167545261, 4.905445897440579, 5.407246019539930, 6.103083756274024, 6.562586641968240, 7.105595062893111, 7.378806384099444, 8.409294639882072, 8.857583705300642, 9.366251916891545, 9.671813855980365, 10.41356381564899, 10.79713849579322, 11.42184909359489, 11.72492899113354, 12.38269176471765, 12.95627017034955, 13.26568208360352, 14.03201879909776, 14.55494843360745
