| L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.164 + 1.72i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (−0.688 − 1.58i)6-s + (−0.487 − 4.63i)7-s + (−0.587 + 0.809i)8-s + (−2.94 + 0.566i)9-s + (−0.669 + 0.743i)10-s + (0.793 + 0.353i)11-s + (1.14 + 1.29i)12-s + (0.254 + 1.19i)13-s + (1.89 + 4.25i)14-s + (1.00 + 1.41i)15-s + (0.309 − 0.951i)16-s + (2.33 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.0947 + 0.995i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (−0.281 − 0.648i)6-s + (−0.184 − 1.75i)7-s + (−0.207 + 0.286i)8-s + (−0.982 + 0.188i)9-s + (−0.211 + 0.235i)10-s + (0.239 + 0.106i)11-s + (0.330 + 0.374i)12-s + (0.0706 + 0.332i)13-s + (0.506 + 1.13i)14-s + (0.259 + 0.364i)15-s + (0.0772 − 0.237i)16-s + (0.567 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647270 - 0.455763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647270 - 0.455763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.164 - 1.72i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (4.35 + 3.47i)T \) |
| good | 7 | \( 1 + (0.487 + 4.63i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.793 - 0.353i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.254 - 1.19i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.33 + 1.04i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (6.87 + 1.46i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (3.09 + 2.24i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.73 + 5.33i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (6.74 + 3.89i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.65 - 3.28i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.00 + 9.43i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-8.17 - 2.65i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.0119 + 0.113i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.95 + 7.16i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 9.88iT - 61T^{2} \) |
| 67 | \( 1 + (0.378 + 0.655i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.04 + 0.215i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-5.12 + 11.5i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.61 - 10.3i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.31 - 2.57i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (5.10 - 3.70i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.19 + 6.68i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.889571315959488133740198719021, −9.239534893719836072887047168964, −8.369826759396285224592671078624, −7.45654327674746448601244382012, −6.57491879051507126420211041686, −5.61461219244194165718724445579, −4.34028393615900245841400148324, −3.82435192335293330256020668029, −2.19279011418811606607780394595, −0.43862953249239806314804400743,
1.64698076289068006310748491853, 2.42295369961885953509155217794, 3.42899981114423358294915773984, 5.45607678243630872865452641876, 6.03461748404163716268304761619, 6.82721590146185436368978649213, 7.903831748908270061231777969286, 8.692898144843782874426548501312, 9.078162825566281347027297875435, 10.18309002976971570493340418827
