L(s) = 1 | + (−0.205 + 0.149i)2-s + (−0.598 + 1.84i)4-s + (−1.71 + 2.36i)5-s + (2.58 + 0.840i)7-s + (−0.308 − 0.949i)8-s − 0.741i·10-s + (2.71 − 1.89i)11-s + (−1.67 − 2.31i)13-s + (−0.655 + 0.213i)14-s + (−2.92 − 2.12i)16-s + (3.60 + 2.62i)17-s + (−1.81 + 0.590i)19-s + (−3.32 − 4.57i)20-s + (−0.275 + 0.795i)22-s + 0.816i·23-s + ⋯ |
L(s) = 1 | + (−0.145 + 0.105i)2-s + (−0.299 + 0.920i)4-s + (−0.768 + 1.05i)5-s + (0.977 + 0.317i)7-s + (−0.109 − 0.335i)8-s − 0.234i·10-s + (0.820 − 0.572i)11-s + (−0.465 − 0.640i)13-s + (−0.175 + 0.0569i)14-s + (−0.731 − 0.531i)16-s + (0.875 + 0.636i)17-s + (−0.416 + 0.135i)19-s + (−0.743 − 1.02i)20-s + (−0.0586 + 0.169i)22-s + 0.170i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.687189 + 0.519287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.687189 + 0.519287i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.71 + 1.89i)T \) |
good | 2 | \( 1 + (0.205 - 0.149i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (1.71 - 2.36i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.58 - 0.840i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (1.67 + 2.31i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.60 - 2.62i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.81 - 0.590i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 0.816iT - 23T^{2} \) |
| 29 | \( 1 + (-2.95 + 9.07i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.84 + 3.51i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.83 + 5.65i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.60 - 8.02i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 + (7.34 - 2.38i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (6.14 + 8.45i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0887 - 0.0288i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.26 + 1.73i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + (2.12 - 2.91i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.67 + 0.543i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.19 + 5.77i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (6.33 + 4.60i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 2.06iT - 89T^{2} \) |
| 97 | \( 1 + (1.35 - 0.981i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.40936077460191607155057749992, −12.98309137167842042097621470153, −11.76962720370162572739700122497, −11.28030143679656902722606553740, −9.782585117251287036289213473307, −8.153741763040928942506711413803, −7.82422983451134820453201417236, −6.29972392161137907512082870187, −4.32881215120726805005211213193, −3.05119460112927957937182536110,
1.33073915980037750972244826925, 4.40098637983589955664125670651, 5.09048116607980684591611272957, 6.97633879380307730502037059082, 8.382799361408208671313195700504, 9.237659472038260883233850932019, 10.45680196668290874826606157249, 11.68643452874393523122694705168, 12.37189077572108964854651038243, 13.98660828091433267010785619854