| L(s) = 1 | + (−0.764 + 1.18i)2-s + (−0.830 − 1.81i)4-s + (−1.90 + 6.47i)5-s + (−7.74 + 8.93i)7-s + (2.79 + 0.402i)8-s + (−6.25 − 7.21i)10-s + (−1.39 − 2.17i)11-s + (−15.9 − 18.4i)13-s + (−4.70 − 16.0i)14-s + (−2.61 + 3.02i)16-s + (17.1 + 7.83i)17-s + (5.09 + 11.1i)19-s + (13.3 − 1.92i)20-s + 3.65·22-s + (21.7 − 7.49i)23-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (−0.207 − 0.454i)4-s + (−0.380 + 1.29i)5-s + (−1.10 + 1.27i)7-s + (0.349 + 0.0503i)8-s + (−0.625 − 0.721i)10-s + (−0.126 − 0.197i)11-s + (−1.22 − 1.41i)13-s + (−0.336 − 1.14i)14-s + (−0.163 + 0.188i)16-s + (1.00 + 0.460i)17-s + (0.268 + 0.587i)19-s + (0.668 − 0.0961i)20-s + 0.165·22-s + (0.945 − 0.325i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0611662 - 0.0676248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0611662 - 0.0676248i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.764 - 1.18i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-21.7 + 7.49i)T \) |
| good | 5 | \( 1 + (1.90 - 6.47i)T + (-21.0 - 13.5i)T^{2} \) |
| 7 | \( 1 + (7.74 - 8.93i)T + (-6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (1.39 + 2.17i)T + (-50.2 + 110. i)T^{2} \) |
| 13 | \( 1 + (15.9 + 18.4i)T + (-24.0 + 167. i)T^{2} \) |
| 17 | \( 1 + (-17.1 - 7.83i)T + (189. + 218. i)T^{2} \) |
| 19 | \( 1 + (-5.09 - 11.1i)T + (-236. + 272. i)T^{2} \) |
| 29 | \( 1 + (24.1 + 11.0i)T + (550. + 635. i)T^{2} \) |
| 31 | \( 1 + (-5.15 + 35.8i)T + (-922. - 270. i)T^{2} \) |
| 37 | \( 1 + (44.6 - 13.1i)T + (1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (4.50 - 15.3i)T + (-1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (0.512 + 3.56i)T + (-1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 + 79.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-49.7 - 43.0i)T + (399. + 2.78e3i)T^{2} \) |
| 59 | \( 1 + (38.0 - 32.9i)T + (495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-0.442 + 3.08i)T + (-3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (59.6 + 38.3i)T + (1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (55.4 - 86.2i)T + (-2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (-0.255 - 0.560i)T + (-3.48e3 + 4.02e3i)T^{2} \) |
| 79 | \( 1 + (89.4 + 103. i)T + (-888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (8.41 + 28.6i)T + (-5.79e3 + 3.72e3i)T^{2} \) |
| 89 | \( 1 + (139. - 20.1i)T + (7.60e3 - 2.23e3i)T^{2} \) |
| 97 | \( 1 + (-162. - 47.7i)T + (7.91e3 + 5.08e3i)T^{2} \) |
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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
−11.68873184713951905971673050852, −10.33462330481339387473759535306, −10.03804615880694860619479273608, −8.904598529942089985483479468047, −7.80832558940971355362529033743, −7.12124786280861902550309017441, −6.01742152435974751797322784658, −5.41188794143925053691047062534, −3.39542862274795939159524816696, −2.63259941558498195371160662112,
0.04726855293732412897141862997, 1.30332798666816139065936328697, 3.16564633880507869394943035508, 4.29037138695521194718093656454, 5.10543510533100384298334843272, 7.01218165318916612905475990217, 7.42139121021971419186641507670, 8.860744138527708904926757787561, 9.457644910758092737173222438503, 10.15322850336197741882885634884
