| L(s) = 1 | + 10.0·3-s + 10.5·7-s + 74.9·9-s − 38.9·11-s + 68.9·13-s − 65.9·17-s + 49.4·19-s + 106.·21-s + 164.·23-s + 484.·27-s − 170.·29-s − 166.·31-s − 392.·33-s + 384.·37-s + 696.·39-s − 22.8·41-s − 136.·43-s + 307.·47-s − 230.·49-s − 665.·51-s − 222·53-s + 499.·57-s − 522.·59-s + 393.·61-s + 793.·63-s + 476.·67-s + 1.66e3·69-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.571·7-s + 2.77·9-s − 1.06·11-s + 1.47·13-s − 0.940·17-s + 0.597·19-s + 1.11·21-s + 1.49·23-s + 3.45·27-s − 1.09·29-s − 0.964·31-s − 2.07·33-s + 1.70·37-s + 2.85·39-s − 0.0868·41-s − 0.484·43-s + 0.954·47-s − 0.672·49-s − 1.82·51-s − 0.575·53-s + 1.16·57-s − 1.15·59-s + 0.824·61-s + 1.58·63-s + 0.869·67-s + 2.89·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.881190935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.881190935\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 10.0T + 27T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 11 | \( 1 + 38.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 384.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 136.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 222T + 1.48e5T^{2} \) |
| 59 | \( 1 + 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 4.26T + 3.57e5T^{2} \) |
| 73 | \( 1 - 601.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 479.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 635.T + 9.12e5T^{2} \) |
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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
−9.481014955915581178659643724363, −9.009122631576902448353995190829, −8.155788743638553478261533115137, −7.67541862941243193745265920620, −6.70893532387616893823689302909, −5.21968153902149064070740709812, −4.13869338822787779008772778132, −3.25531984698575002359941391728, −2.33465879082082092819998701923, −1.28130439619926516908075037396,
1.28130439619926516908075037396, 2.33465879082082092819998701923, 3.25531984698575002359941391728, 4.13869338822787779008772778132, 5.21968153902149064070740709812, 6.70893532387616893823689302909, 7.67541862941243193745265920620, 8.155788743638553478261533115137, 9.009122631576902448353995190829, 9.481014955915581178659643724363
