| L(s) = 1 | + 2.73·3-s − 4.73·7-s + 4.46·9-s + 3.46·11-s + 3.46·13-s + 3.46·17-s + 2·19-s − 12.9·21-s + 2.19·23-s + 3.99·27-s + 2.53·31-s + 9.46·33-s − 6·37-s + 9.46·39-s − 9.46·41-s + 0.196·43-s + 2.19·47-s + 15.3·49-s + 9.46·51-s − 10.3·53-s + 5.46·57-s + 6·59-s + 0.928·61-s − 21.1·63-s − 0.196·67-s + 6·69-s + 16.3·71-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 1.78·7-s + 1.48·9-s + 1.04·11-s + 0.960·13-s + 0.840·17-s + 0.458·19-s − 2.82·21-s + 0.457·23-s + 0.769·27-s + 0.455·31-s + 1.64·33-s − 0.986·37-s + 1.51·39-s − 1.47·41-s + 0.0299·43-s + 0.320·47-s + 2.19·49-s + 1.32·51-s − 1.42·53-s + 0.723·57-s + 0.781·59-s + 0.118·61-s − 2.66·63-s − 0.0239·67-s + 0.722·69-s + 1.94·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.502263944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.502263944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 0.928T + 61T^{2} \) |
| 67 | \( 1 + 0.196T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.168709701098219202126105374692, −7.35054805015558488062165901132, −6.64391074052725422272919611741, −6.22096642621527984034208576557, −5.12771536831812951744003174210, −3.84908960956245359178633337269, −3.48457856032301481146845050714, −3.08152240082243733422938033957, −1.99483618197493534492317685644, −0.924739637637147145409087689037,
0.924739637637147145409087689037, 1.99483618197493534492317685644, 3.08152240082243733422938033957, 3.48457856032301481146845050714, 3.84908960956245359178633337269, 5.12771536831812951744003174210, 6.22096642621527984034208576557, 6.64391074052725422272919611741, 7.35054805015558488062165901132, 8.168709701098219202126105374692
