| L(s) = 1 | + (−3.57 − 0.566i)2-s + (1.61 + 3.17i)3-s + (8.65 + 2.81i)4-s + (−0.872 + 4.92i)5-s + (−3.98 − 12.2i)6-s + (−0.574 − 0.574i)7-s + (−16.4 − 8.37i)8-s + (−2.16 + 2.97i)9-s + (5.90 − 17.1i)10-s + (1.94 − 1.41i)11-s + (5.06 + 31.9i)12-s + (7.34 − 1.16i)13-s + (1.72 + 2.37i)14-s + (−17.0 + 5.19i)15-s + (24.5 + 17.8i)16-s + (12.9 − 25.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 − 0.283i)2-s + (0.538 + 1.05i)3-s + (2.16 + 0.702i)4-s + (−0.174 + 0.984i)5-s + (−0.663 − 2.04i)6-s + (−0.0820 − 0.0820i)7-s + (−2.05 − 1.04i)8-s + (−0.240 + 0.331i)9-s + (0.590 − 1.71i)10-s + (0.176 − 0.128i)11-s + (0.422 + 2.66i)12-s + (0.564 − 0.0894i)13-s + (0.123 + 0.169i)14-s + (−1.13 + 0.346i)15-s + (1.53 + 1.11i)16-s + (0.761 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.452262 + 0.256441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452262 + 0.256441i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.872 - 4.92i)T \) |
| good | 2 | \( 1 + (3.57 + 0.566i)T + (3.80 + 1.23i)T^{2} \) |
| 3 | \( 1 + (-1.61 - 3.17i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (0.574 + 0.574i)T + 49iT^{2} \) |
| 11 | \( 1 + (-1.94 + 1.41i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-7.34 + 1.16i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-12.9 + 25.3i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (9.67 - 3.14i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (0.567 - 3.58i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-25.0 - 8.14i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-5.05 - 15.5i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (5.67 + 35.8i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (49.2 + 35.7i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (30.0 - 30.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (55.2 - 28.1i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-24.7 - 48.4i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-45.2 + 62.3i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-0.372 + 0.270i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (22.5 - 44.2i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (22.1 - 68.1i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-3.89 + 24.6i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (15.5 + 5.06i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (93.7 + 47.7i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-2.75 - 3.79i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-101. + 51.5i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−17.84802042575090284680260097169, −16.34991276773000263624478159292, −15.63000856248323680267632233120, −14.32915621313204292216319819549, −11.68458816819184731244036687401, −10.50611855594946723225301790813, −9.725352538746735911629160879311, −8.482081227301990879438781671975, −6.96214169276506491048777292422, −3.17292503335480803790711307968,
1.50632203761610622419326829066, 6.49268367883661067227919148021, 8.057985020049639672492005969807, 8.582924208044083987531494653495, 10.14924749036032493973762084576, 11.92197245766068273971308470229, 13.22680348176014728234996642711, 15.12424934851876375934348128356, 16.45604742914015849481776497273, 17.31112994771102984766582528486
