| L(s) = 1 | − 2.14·3-s + 3.14·5-s − 1.74·7-s + 1.60·9-s − 2.89·11-s + 2.39·13-s − 6.74·15-s − 5.49·17-s + 2·19-s + 3.74·21-s + 1.60·23-s + 4.89·25-s + 3.00·27-s + 5.89·29-s − 3.94·31-s + 6.20·33-s − 5.49·35-s + 37-s − 5.14·39-s + 6.14·41-s + 5.20·43-s + 5.03·45-s − 0.253·47-s − 3.94·49-s + 11.7·51-s − 0.543·53-s − 9.09·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 1.40·5-s − 0.660·7-s + 0.533·9-s − 0.871·11-s + 0.665·13-s − 1.74·15-s − 1.33·17-s + 0.458·19-s + 0.817·21-s + 0.333·23-s + 0.978·25-s + 0.577·27-s + 1.09·29-s − 0.708·31-s + 1.07·33-s − 0.928·35-s + 0.164·37-s − 0.823·39-s + 0.959·41-s + 0.793·43-s + 0.750·45-s − 0.0369·47-s − 0.564·49-s + 1.64·51-s − 0.0746·53-s − 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.176039608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176039608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 41 | \( 1 - 6.14T + 41T^{2} \) |
| 43 | \( 1 - 5.20T + 43T^{2} \) |
| 47 | \( 1 + 0.253T + 47T^{2} \) |
| 53 | \( 1 + 0.543T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 - 7.14T + 67T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 9.89T + 79T^{2} \) |
| 83 | \( 1 - 6.03T + 83T^{2} \) |
| 89 | \( 1 - 4.50T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.186591344597746103100171410353, −8.320093960243542889226624321840, −7.06317299565379874992181075550, −6.42646520131469217312261329823, −5.87019981870770555938345303439, −5.29560530306203953859307446414, −4.44946878594578753099652007132, −3.03148256656862469094711242963, −2.09901471378667170854139393802, −0.72947016341849497422566936636,
0.72947016341849497422566936636, 2.09901471378667170854139393802, 3.03148256656862469094711242963, 4.44946878594578753099652007132, 5.29560530306203953859307446414, 5.87019981870770555938345303439, 6.42646520131469217312261329823, 7.06317299565379874992181075550, 8.320093960243542889226624321840, 9.186591344597746103100171410353
