L(s) = 1 | + i·2-s + (1.66 + 0.462i)3-s − 4-s − 5-s + (−0.462 + 1.66i)6-s − 2.82i·7-s − i·8-s + (2.57 + 1.54i)9-s − i·10-s + 0.0884·11-s + (−1.66 − 0.462i)12-s + 5.38·13-s + 2.82·14-s + (−1.66 − 0.462i)15-s + 16-s + 2.84·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.963 + 0.267i)3-s − 0.5·4-s − 0.447·5-s + (−0.188 + 0.681i)6-s − 1.06i·7-s − 0.353i·8-s + (0.857 + 0.514i)9-s − 0.316i·10-s + 0.0266·11-s + (−0.481 − 0.133i)12-s + 1.49·13-s + 0.754·14-s + (−0.430 − 0.119i)15-s + 0.250·16-s + 0.689·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79679 + 0.894566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79679 + 0.894566i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.66 - 0.462i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-4.45 - 1.76i)T \) |
good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 0.0884T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 - 5.08iT - 19T^{2} \) |
| 29 | \( 1 - 1.48iT - 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 6.20iT - 37T^{2} \) |
| 41 | \( 1 + 0.519iT - 41T^{2} \) |
| 43 | \( 1 - 0.907iT - 43T^{2} \) |
| 47 | \( 1 + 0.0243iT - 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 8.85iT - 59T^{2} \) |
| 61 | \( 1 + 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 3.80iT - 67T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 - 9.17iT - 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 7.83iT - 97T^{2} \) |
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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
−10.50469465760969909543474285659, −9.534895230997316493696854242255, −8.724492737376969066606774673051, −7.87123109328017672362352938106, −7.39240224672077582994944571293, −6.31805698866969969299116631331, −5.06474846776601093773927391355, −3.81149049871236974860754773819, −3.51688354680840495885017749817, −1.35371379130459133231516563495,
1.30303788680647710818136581720, 2.67880105240889106641849654961, 3.41178504014738955848754811914, 4.53777552653654070729587758085, 5.80247554979908495668207416550, 6.96229524492367132758937608838, 8.058147943084779847669790778890, 8.830404362703388673523867386315, 9.176132291324521761703358542839, 10.36088114385313099648445657865