| L(s) = 1 | + (−2.18 + 1.59i)2-s + (0.309 − 0.951i)3-s + (1.64 − 5.06i)4-s + (0.336 − 2.21i)5-s + (0.836 + 2.57i)6-s + 0.470·7-s + (2.78 + 8.55i)8-s + (−0.809 − 0.587i)9-s + (2.78 + 5.37i)10-s + (2.57 − 1.87i)11-s + (−4.30 − 3.12i)12-s + (−0.455 − 0.331i)13-s + (−1.02 + 0.748i)14-s + (−1.99 − 1.00i)15-s + (−11.0 − 8.05i)16-s + (−0.527 − 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 1.12i)2-s + (0.178 − 0.549i)3-s + (0.822 − 2.53i)4-s + (0.150 − 0.988i)5-s + (0.341 + 1.05i)6-s + 0.177·7-s + (0.982 + 3.02i)8-s + (−0.269 − 0.195i)9-s + (0.879 + 1.69i)10-s + (0.776 − 0.563i)11-s + (−1.24 − 0.903i)12-s + (−0.126 − 0.0918i)13-s + (−0.275 + 0.199i)14-s + (−0.516 − 0.258i)15-s + (−2.77 − 2.01i)16-s + (−0.127 − 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516036 - 0.0456038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516036 - 0.0456038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.336 + 2.21i)T \) |
| good | 2 | \( 1 + (2.18 - 1.59i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 - 0.470T + 7T^{2} \) |
| 11 | \( 1 + (-2.57 + 1.87i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.455 + 0.331i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.527 + 1.62i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.15 - 3.55i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.83 + 1.33i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.57 - 7.91i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.67 - 5.17i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.825 + 0.600i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.19 + 0.865i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + (1.37 - 4.21i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.17 - 6.70i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 7.56i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.102 - 0.0745i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.863 + 2.65i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.95 + 15.2i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.91 + 5.75i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.46 - 4.52i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.61 - 11.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.01 + 3.64i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.61 + 8.04i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.70265289186354281720963673258, −13.86477892528609217399001130322, −12.24472206524706177516929189165, −10.85949819132874383355828628689, −9.420832140826529847468994114061, −8.734334656031553475303353197609, −7.77444727406043832790750618608, −6.55204703138270377280349323383, −5.32741251771318038482834534986, −1.30193379590582868871331673828,
2.33206772867143503342010671730, 3.83999394468772514139806867964, 6.82897089154161743614652082254, 8.026561762881620037922145775041, 9.382472355091529155670027352239, 9.964563366249223358187566048381, 11.13208719026265471666077291270, 11.69307309142033945433954130681, 13.22616322751963732878220003854, 14.76359870996940172127335376877
