| L(s) = 1 | + (−0.250 + 1.17i)2-s + (0.502 + 0.223i)4-s + (1.04 − 1.97i)5-s + (2.32 + 1.33i)7-s + (−1.80 + 2.48i)8-s + (2.06 + 1.72i)10-s + (−1.37 − 0.292i)11-s + (−0.0338 − 0.159i)13-s + (−2.15 + 2.39i)14-s + (−1.73 − 1.93i)16-s + (−0.113 + 0.156i)17-s + (6.07 + 4.41i)19-s + (0.966 − 0.760i)20-s + (0.688 − 1.54i)22-s + (4.38 + 3.94i)23-s + ⋯ |
| L(s) = 1 | + (−0.177 + 0.832i)2-s + (0.251 + 0.111i)4-s + (0.466 − 0.884i)5-s + (0.877 + 0.506i)7-s + (−0.638 + 0.878i)8-s + (0.654 + 0.545i)10-s + (−0.414 − 0.0881i)11-s + (−0.00937 − 0.0441i)13-s + (−0.577 + 0.640i)14-s + (−0.434 − 0.482i)16-s + (−0.0275 + 0.0378i)17-s + (1.39 + 1.01i)19-s + (0.216 − 0.169i)20-s + (0.146 − 0.329i)22-s + (0.914 + 0.823i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43423 + 1.08690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43423 + 1.08690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.04 + 1.97i)T \) |
| good | 2 | \( 1 + (0.250 - 1.17i)T + (-1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 1.33i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.37 + 0.292i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.0338 + 0.159i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.113 - 0.156i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-6.07 - 4.41i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.38 - 3.94i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.636 + 6.06i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.0629 + 0.598i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.888 - 0.288i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.53 + 1.17i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-5.91 - 3.41i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.217 + 0.0228i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-7.64 - 10.5i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (5.81 - 1.23i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (13.7 + 2.93i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (6.42 + 0.675i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (2.90 - 2.11i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (11.0 - 3.59i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.42 + 13.5i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.96 + 13.3i)T + (-55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.38 - 10.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.08 + 0.323i)T + (94.8 - 20.1i)T^{2} \) |
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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
−10.67939903893939656313312550324, −9.466856056444829037432933481868, −8.845094419943833157429749764876, −7.85202153977606918439666423189, −7.51276322114208292203967434259, −5.90011926139248987491891050752, −5.61724044166858740653380780141, −4.56060650179686395063668448159, −2.85099912534759628385197261228, −1.54155709764528253558505623341,
1.18580694355499230716623479108, 2.46784196243333890052107453775, 3.27105733739256564114648194827, 4.72492925516653828795615272610, 5.82170655651886407062928750913, 7.01303779094572770231947679866, 7.42599675029324247043326213833, 8.888520354873918622227269676811, 9.708811145661305288857623110932, 10.62776176814101077105101435769
