L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.499 − 0.866i)6-s + (1 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−1 + 1.73i)11-s − 0.999·12-s + (2.5 + 2.59i)13-s + 1.99·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−2.5 − 4.33i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.204 − 0.353i)6-s + (0.377 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.301 + 0.522i)11-s − 0.288·12-s + (0.693 + 0.720i)13-s + 0.534·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.956349 - 0.568623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.956349 - 0.568623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-2.5 - 2.59i)T \) |
good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.02284794869150380654077828001, −13.22864199670998726249811629288, −11.87481719325640869891287111228, −11.50830975629029436416191828239, −9.813277683056482657183578413109, −8.651546379069765902010011758834, −7.35495711273524785367998704520, −5.71353738541044452852246427228, −4.06164971991764944799633028143, −2.19914849365351686454379623697,
3.49199774966556626951947285055, 4.74671246253920085262181588315, 6.31853884050067719990124591716, 7.892840979726185351712776661992, 8.606747222634992294453947391565, 10.32309361018841613699534679725, 11.24536489118274434866906356977, 12.78797333273494567495868239787, 13.74556078736786250052476462286, 14.72335395250540385418313040984