| 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
−11.56738997697051011606428115284, −10.53789001842868626659561301315, −9.532672145774340239691343654926, −8.835609351522906087957626827584, −7.46832072610149796713753973754, −6.66316980370285431611096368681, −5.28136498349557179441767659792, −4.54336634275867436610717101716, −3.00943220754072508205102632688, −2.20024352431318745740436581391,
1.17921448290620566959215892189, 3.02776310722934582923420773692, 4.58382788041239992241516300878, 5.43366493442251769542124773850, 6.09458651965138779794414717970, 7.21591413529217253580804091607, 8.398646395438730572100019533808, 9.536589640838619372751374684035, 10.15815700553343106979694071640, 10.92407568586431660469202045297
