| L(s) = 1 | + 2.36·3-s + 1.03·5-s + 5.11·7-s + 2.59·9-s + 4.82·11-s − 5.73·13-s + 2.45·15-s + 2·17-s + 6.54·19-s + 12.1·21-s − 0.940·23-s − 3.91·25-s − 0.962·27-s − 5.73·29-s − 1.51·31-s + 11.4·33-s + 5.31·35-s − 37-s − 13.5·39-s − 2.36·41-s − 8.42·43-s + 2.69·45-s + 0.200·47-s + 19.1·49-s + 4.72·51-s − 13.2·53-s + 5.01·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 0.464·5-s + 1.93·7-s + 0.864·9-s + 1.45·11-s − 1.59·13-s + 0.634·15-s + 0.485·17-s + 1.50·19-s + 2.64·21-s − 0.196·23-s − 0.783·25-s − 0.185·27-s − 1.06·29-s − 0.272·31-s + 1.98·33-s + 0.898·35-s − 0.164·37-s − 2.17·39-s − 0.369·41-s − 1.28·43-s + 0.401·45-s + 0.0293·47-s + 2.73·49-s + 0.662·51-s − 1.82·53-s + 0.675·55-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
| \(L(1)\) |
\(\approx\) |
\(4.052941388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.052941388\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.36T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 0.940T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 - 0.200T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 + 5.03T + 61T^{2} \) |
| 67 | \( 1 + 8.64T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 + 0.940T + 79T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 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
−9.037374389573868333395336262278, −8.109788484480737177180476105427, −7.66721123406205727275941883732, −7.03533801409517292443674906180, −5.64251944180275113312167315814, −4.95187189057291902387594531149, −4.06476087913704071622313298148, −3.13499334839139048499377352713, −1.97321339470979449287030318253, −1.52891285973190667298374828210,
1.52891285973190667298374828210, 1.97321339470979449287030318253, 3.13499334839139048499377352713, 4.06476087913704071622313298148, 4.95187189057291902387594531149, 5.64251944180275113312167315814, 7.03533801409517292443674906180, 7.66721123406205727275941883732, 8.109788484480737177180476105427, 9.037374389573868333395336262278
