| 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
| \(L(2)\) |
\(\approx\) |
\(0.01402728934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01402728934\) |
| \(L(\frac{5}{2})\) |
|
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
−8.915723273373778835269994371599, −8.380141676727533511000602030583, −8.020053755613851084485021948891, −7.13988732184619178975354622754, −6.42198357112769872567442688951, −5.03302943880914697637428259475, −3.94879913533536718407532391964, −2.80673665770956677217644193191, −1.64727491552508987929288088891, −0.07471588277330571986434297865,
0.07471588277330571986434297865, 1.64727491552508987929288088891, 2.80673665770956677217644193191, 3.94879913533536718407532391964, 5.03302943880914697637428259475, 6.42198357112769872567442688951, 7.13988732184619178975354622754, 8.020053755613851084485021948891, 8.380141676727533511000602030583, 8.915723273373778835269994371599
