| L(s) = 1 | + (−1.09 + 0.322i)2-s + (−0.654 − 0.755i)3-s + (−0.581 + 0.373i)4-s + (0.142 + 0.989i)5-s + (0.962 + 0.618i)6-s + (−0.437 + 0.957i)7-s + (2.01 − 2.32i)8-s + (−0.142 + 0.989i)9-s + (−0.475 − 1.04i)10-s + (−0.593 − 0.174i)11-s + (0.663 + 0.194i)12-s + (−0.220 − 0.482i)13-s + (0.171 − 1.19i)14-s + (0.654 − 0.755i)15-s + (−0.888 + 1.94i)16-s + (−5.81 − 3.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.776 + 0.227i)2-s + (−0.378 − 0.436i)3-s + (−0.290 + 0.186i)4-s + (0.0636 + 0.442i)5-s + (0.392 + 0.252i)6-s + (−0.165 + 0.361i)7-s + (0.712 − 0.822i)8-s + (−0.0474 + 0.329i)9-s + (−0.150 − 0.329i)10-s + (−0.179 − 0.0525i)11-s + (0.191 + 0.0562i)12-s + (−0.0611 − 0.133i)13-s + (0.0457 − 0.318i)14-s + (0.169 − 0.195i)15-s + (−0.222 + 0.486i)16-s + (−1.40 − 0.906i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00982121 - 0.0427277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00982121 - 0.0427277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.460 + 4.77i)T \) |
| good | 2 | \( 1 + (1.09 - 0.322i)T + (1.68 - 1.08i)T^{2} \) |
| 7 | \( 1 + (0.437 - 0.957i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.593 + 0.174i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.220 + 0.482i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.81 + 3.73i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.95 - 2.53i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (7.10 + 4.56i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (5.94 - 6.85i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.202 - 1.40i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.899 + 6.25i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (6.59 + 7.60i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 + (-4.37 + 9.57i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-5.63 - 12.3i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.60 + 4.16i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.44 + 1.30i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.69 - 2.84i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (2.65 - 1.70i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 4.31i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.10 - 7.70i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.869 - 1.00i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.13 - 7.89i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.91834740957539735578553286216, −10.19865942114384493954162462301, −9.059687599422270478107275695479, −8.419323096604847796626741216066, −7.24563587612800389080831987891, −6.63204376048360881921012729244, −5.30284405206930360124598094289, −3.95241246895096648248716425459, −2.23335144190814350553684068857, −0.03996353883919824420857285531,
1.84159658482065159084254460738, 3.97528116746560998435601412651, 4.89701298529259246737415978418, 6.00717485582921226970197611165, 7.32569899093853450858413463291, 8.531025938226407910462940043579, 9.199089811283028771671642843289, 9.983195957767739707936893008438, 10.94266761588586517304346047203, 11.38884686841747012516419131130
