| L(s) = 1 | + (0.309 − 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (−0.272 + 2.59i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (−4.40 + 1.96i)11-s + (−0.978 + 0.207i)12-s + (−1.10 − 0.234i)13-s + (2.38 + 1.05i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−4.16 − 1.85i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.386 − 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.102 + 0.979i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (0.211 + 0.235i)10-s + (−1.32 + 0.591i)11-s + (−0.282 + 0.0600i)12-s + (−0.306 − 0.0651i)13-s + (0.636 + 0.283i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.01 − 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.265999 + 0.335522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.265999 + 0.335522i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (3.55 - 4.28i)T \) |
| good | 7 | \( 1 + (0.272 - 2.59i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (4.40 - 1.96i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (1.10 + 0.234i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (4.16 + 1.85i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (3.63 - 0.772i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (3.62 - 2.63i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.109 + 0.336i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (2.47 + 4.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.05 + 4.50i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-8.37 + 1.78i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.56 - 4.82i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.656 - 6.24i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.787 + 0.874i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 + (0.256 - 0.443i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.17 + 11.1i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.837 - 0.372i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-7.45 - 3.32i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.59 - 7.32i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-3.79 - 2.76i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (14.6 + 10.6i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.46439769725114030576199062424, −9.454337461870632696822486064848, −8.755905563720349668104623841722, −7.84247181248208105203625196686, −7.00325210904765698291380649200, −5.86993127335072038580577923694, −5.00405352065432431632356368223, −3.83400788826152061378865822066, −2.56265429496944074996858861062, −2.15232031383639437977972538417,
0.16186068499868839287048983757, 2.41696646529155615862412602094, 3.77309208468352709626442398612, 4.44483524141281322739656328982, 5.35246144374071659661426971358, 6.43617070580092175583000436045, 7.38046986292630437201764586694, 8.205506412962323857321915979183, 8.684214276133851378373826909021, 9.837676914372714955149051009303
