| L(s) = 1 | + 4.67·2-s + 13.8·4-s + 15.5·5-s + 27.2·8-s + 72.6·10-s + 63.4·11-s + 53.1·13-s + 16.7·16-s + 68.4·17-s − 86.0·19-s + 215.·20-s + 296.·22-s + 46.9·23-s + 116.·25-s + 248.·26-s − 169.·29-s − 141.·31-s − 139.·32-s + 319.·34-s − 411.·37-s − 402.·38-s + 423.·40-s + 49.0·41-s − 356.·43-s + 878.·44-s + 219.·46-s + 387.·47-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.72·4-s + 1.39·5-s + 1.20·8-s + 2.29·10-s + 1.74·11-s + 1.13·13-s + 0.261·16-s + 0.976·17-s − 1.03·19-s + 2.40·20-s + 2.87·22-s + 0.425·23-s + 0.932·25-s + 1.87·26-s − 1.08·29-s − 0.821·31-s − 0.772·32-s + 1.61·34-s − 1.83·37-s − 1.71·38-s + 1.67·40-s + 0.186·41-s − 1.26·43-s + 3.00·44-s + 0.702·46-s + 1.20·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.925877451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.925877451\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.67T + 8T^{2} \) |
| 5 | \( 1 - 15.5T + 125T^{2} \) |
| 11 | \( 1 - 63.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 411.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 49.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 821.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 95.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 750.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 23.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 592.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 864.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 614.T + 9.12e5T^{2} \) |
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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
−9.242056754993222261962604439377, −8.635242014122291341243727678232, −7.02119501166248056716636050174, −6.47786500653590877605636292428, −5.78149100928589830391819239007, −5.22720882022483938817046143498, −3.94497462248499890989606755249, −3.49611404819188545812844493654, −2.11059585158939446179143385588, −1.38860969684012377389721046098,
1.38860969684012377389721046098, 2.11059585158939446179143385588, 3.49611404819188545812844493654, 3.94497462248499890989606755249, 5.22720882022483938817046143498, 5.78149100928589830391819239007, 6.47786500653590877605636292428, 7.02119501166248056716636050174, 8.635242014122291341243727678232, 9.242056754993222261962604439377
