| L(s) = 1 | + (0.807 + 0.565i)2-s + (−0.351 − 0.966i)4-s + (2.04 − 0.911i)5-s + (−0.275 − 0.590i)7-s + (0.772 − 2.88i)8-s + (2.16 + 0.418i)10-s + (−0.890 − 1.06i)11-s + (−2.93 − 4.19i)13-s + (0.111 − 0.632i)14-s + (0.677 − 0.568i)16-s + (1.18 + 4.41i)17-s + (0.00652 − 0.00376i)19-s + (−1.59 − 1.65i)20-s + (−0.118 − 1.36i)22-s + (6.70 + 3.12i)23-s + ⋯ |
| L(s) = 1 | + (0.570 + 0.399i)2-s + (−0.175 − 0.483i)4-s + (0.913 − 0.407i)5-s + (−0.104 − 0.223i)7-s + (0.273 − 1.01i)8-s + (0.684 + 0.132i)10-s + (−0.268 − 0.319i)11-s + (−0.814 − 1.16i)13-s + (0.0298 − 0.169i)14-s + (0.169 − 0.142i)16-s + (0.286 + 1.06i)17-s + (0.00149 − 0.000864i)19-s + (−0.357 − 0.369i)20-s + (−0.0253 − 0.289i)22-s + (1.39 + 0.651i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80543 - 0.641221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80543 - 0.641221i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.04 + 0.911i)T \) |
| good | 2 | \( 1 + (-0.807 - 0.565i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (0.275 + 0.590i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.890 + 1.06i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.93 + 4.19i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.18 - 4.41i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.00652 + 0.00376i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.70 - 3.12i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.586 - 3.32i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 1.35i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.60 + 0.430i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.90 - 0.335i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.929 - 10.6i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (6.61 - 3.08i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-1.18 - 1.18i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.77 - 6.52i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.02 + 3.28i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (12.3 - 8.62i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.41 + 3.12i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 3.25i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.782 - 0.137i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.32 - 7.59i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (5.89 + 10.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.18 + 0.190i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.92407568586431660469202045297, −10.15815700553343106979694071640, −9.536589640838619372751374684035, −8.398646395438730572100019533808, −7.21591413529217253580804091607, −6.09458651965138779794414717970, −5.43366493442251769542124773850, −4.58382788041239992241516300878, −3.02776310722934582923420773692, −1.17921448290620566959215892189,
2.20024352431318745740436581391, 3.00943220754072508205102632688, 4.54336634275867436610717101716, 5.28136498349557179441767659792, 6.66316980370285431611096368681, 7.46832072610149796713753973754, 8.835609351522906087957626827584, 9.532672145774340239691343654926, 10.53789001842868626659561301315, 11.56738997697051011606428115284
