| L(s) = 1 | − 2.58·3-s − 5-s + 3.67·9-s + 1.67·11-s + 4.84·13-s + 2.58·15-s − 2·17-s − 6.84·19-s − 2.26·23-s + 25-s − 1.75·27-s + 3.32·29-s − 9.16·31-s − 4.33·33-s − 2.84·37-s − 12.5·39-s − 9.52·41-s + 6.58·43-s − 3.67·45-s + 12.2·47-s + 5.16·51-s + 7.49·53-s − 1.67·55-s + 17.6·57-s − 8·59-s + 6.49·61-s − 4.84·65-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 0.447·5-s + 1.22·9-s + 0.506·11-s + 1.34·13-s + 0.667·15-s − 0.485·17-s − 1.57·19-s − 0.471·23-s + 0.200·25-s − 0.337·27-s + 0.616·29-s − 1.64·31-s − 0.754·33-s − 0.467·37-s − 2.00·39-s − 1.48·41-s + 1.00·43-s − 0.548·45-s + 1.78·47-s + 0.723·51-s + 1.02·53-s − 0.226·55-s + 2.34·57-s − 1.04·59-s + 0.830·61-s − 0.601·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7770392680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7770392680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 + 2.84T + 37T^{2} \) |
| 41 | \( 1 + 9.52T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 7.49T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.013527449236951964145087425168, −8.551717851379427426580032622484, −7.41088780443883984885005928262, −6.53199870796999551881665055607, −6.13492072568219315861773843817, −5.25417497271019578441444863586, −4.28680710612776025734278393241, −3.66923165056129107685871443085, −1.94629881718467068253807104436, −0.63655274636668316251165515303,
0.63655274636668316251165515303, 1.94629881718467068253807104436, 3.66923165056129107685871443085, 4.28680710612776025734278393241, 5.25417497271019578441444863586, 6.13492072568219315861773843817, 6.53199870796999551881665055607, 7.41088780443883984885005928262, 8.551717851379427426580032622484, 9.013527449236951964145087425168
