| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (1.68 − 3.68i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (3.62 + 1.06i)11-s + (0.959 + 0.281i)12-s + (−0.305 − 0.669i)13-s + (0.576 − 4.00i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (−2.85 − 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (0.343 + 0.220i)6-s + (0.635 − 1.39i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (1.09 + 0.321i)11-s + (0.276 + 0.0813i)12-s + (−0.0848 − 0.185i)13-s + (0.154 − 1.07i)14-s + (−0.169 + 0.195i)15-s + (0.103 − 0.227i)16-s + (−0.693 − 0.445i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.78335 - 0.246805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.78335 - 0.246805i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.04 - 4.34i)T \) |
| good | 7 | \( 1 + (-1.68 + 3.68i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.62 - 1.06i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.305 + 0.669i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.85 + 1.83i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.838 - 0.538i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.54 - 1.63i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.94 + 5.70i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.363 - 2.52i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.274 - 1.90i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.0781 + 0.0902i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 + (0.466 - 1.02i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.44 - 9.73i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (4.96 - 5.72i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (9.77 - 2.86i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (7.42 - 2.18i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (12.1 - 7.83i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.110 - 0.242i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.73 + 12.0i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (10.8 + 12.5i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.48 + 17.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.40264804471969338688688197117, −9.912427881168499760254849871392, −8.785633361238817138385192193777, −7.59057839704792104698316937801, −6.99196548386180372490607346962, −5.91053476179651032129323756116, −4.46440696120516878789903532872, −4.15300196464143428604514896566, −2.93674589149833222435826369165, −1.47896232217870134478114400961,
1.68716201632027960473231173253, 2.69303486068026262746520695893, 4.09492335877377582346595111623, 5.02260680152170203472146501286, 6.10572905797295170782633269487, 6.67715633263521458252489155363, 8.065045394811124816118954402648, 8.654881857441056560657100209957, 9.265148505117529536249561205302, 10.69508097057545726795102555560
