L(s) = 1 | + (−1.51 − 1.51i)2-s + (0.852 − 3.18i)3-s + 2.59i·4-s + (−1.57 + 1.58i)5-s + (−6.11 + 3.53i)6-s + (0.903 − 3.37i)7-s + (0.909 − 0.909i)8-s + (−6.78 − 3.91i)9-s + (4.79 − 0.00681i)10-s + (2.73 + 1.58i)11-s + (8.26 + 2.21i)12-s + (−2.14 + 0.573i)13-s + (−6.48 + 3.74i)14-s + (3.68 + 6.37i)15-s + 2.44·16-s + (2.29 + 0.615i)17-s + ⋯ |
L(s) = 1 | + (−1.07 − 1.07i)2-s + (0.491 − 1.83i)3-s + 1.29i·4-s + (−0.706 + 0.708i)5-s + (−2.49 + 1.44i)6-s + (0.341 − 1.27i)7-s + (0.321 − 0.321i)8-s + (−2.26 − 1.30i)9-s + (1.51 − 0.00215i)10-s + (0.825 + 0.476i)11-s + (2.38 + 0.639i)12-s + (−0.594 + 0.159i)13-s + (−1.73 + 0.999i)14-s + (0.952 + 1.64i)15-s + 0.610·16-s + (0.557 + 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.130521 + 0.614000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.130521 + 0.614000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.57 - 1.58i)T \) |
| 31 | \( 1 + (-2.33 + 5.05i)T \) |
good | 2 | \( 1 + (1.51 + 1.51i)T + 2iT^{2} \) |
| 3 | \( 1 + (-0.852 + 3.18i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.903 + 3.37i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 1.58i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.14 - 0.573i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.29 - 0.615i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.05 + 1.18i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.57 + 1.57i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 37 | \( 1 + (-3.52 - 0.943i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.995 + 1.72i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.494 + 1.84i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (2.97 + 2.97i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.25 - 0.336i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.708i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 5.71iT - 61T^{2} \) |
| 67 | \( 1 + (-7.80 + 2.09i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.08 + 1.88i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.40 - 0.913i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-16.7 + 4.49i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + (-8.87 - 8.87i)T + 97iT^{2} \) |
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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
−11.96530639082390507528585382125, −11.63999138583194923454298768142, −10.48899258820904233838438485727, −9.285344543471263385368639377757, −7.943491266116722822966285407106, −7.55091671764308142796767120412, −6.58128740515189894084601418305, −3.64886643862653503804085652262, −2.26494056089094817458030014856, −0.854499234937580292002424850659,
3.41295228035458571050565790011, 4.91868742947584353982469053515, 5.80684271651218663952304308301, 7.84354180779333602812145704206, 8.560401519792843286412810608151, 9.234048774537627516698439312249, 9.843677287571665397584968490727, 11.27072399533020302251244834417, 12.19160909897523531106437436567, 14.28819348701847689422080040035