| 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 + (−1.24 − 2.72i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.415 + 0.909i)10-s + (−2.90 + 0.852i)11-s + (−0.959 + 0.281i)12-s + (−2.32 + 5.08i)13-s + (0.425 + 2.96i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (2.15 − 1.38i)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 + (−0.469 − 1.02i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.131 + 0.287i)10-s + (−0.875 + 0.256i)11-s + (−0.276 + 0.0813i)12-s + (−0.644 + 1.41i)13-s + (0.113 + 0.791i)14-s + (0.169 + 0.195i)15-s + (0.103 + 0.227i)16-s + (0.522 − 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.255962 + 0.348204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255962 + 0.348204i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.04 - 2.56i)T \) |
| good | 7 | \( 1 + (1.24 + 2.72i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.90 - 0.852i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.32 - 5.08i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.15 + 1.38i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.20 - 2.70i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.575 - 0.369i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.81 - 5.55i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.504 - 3.50i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.414 - 2.88i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (6.05 - 6.99i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 + (2.27 + 4.98i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (5.20 - 11.4i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.84 - 10.2i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.15 + 0.632i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (6.54 + 1.92i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.45 - 1.57i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.28 + 5.00i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.408 + 2.83i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.92 + 2.22i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.06 + 7.39i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.32448160891844218173881215994, −9.977536465121844320320208539361, −9.330928835472122480410374446091, −8.116107374438005356441479536541, −7.33169026140062153086296576358, −6.47287951417558432501136195606, −5.19978491363068411213729266367, −4.26962728375842284048135376537, −3.09791196262011265148873827446, −1.41357295767931540824815225854,
0.30153751580928731043466560585, 2.30609512678866969139261479966, 3.12427718685936586183511657408, 5.23257208319256870549316521748, 5.77402754402811598760689492001, 6.69874876648152040597951073697, 7.79856020931427711605331304336, 8.202098653825315988543863202979, 9.501694440134936039499236231446, 10.13132495505144624181904259953
