| L(s) = 1 | − 2.20·3-s + (1.71 − 1.43i)5-s + 0.642i·7-s + 1.85·9-s − 2.87·11-s + 0.405i·13-s + (−3.78 + 3.15i)15-s + 4.93·17-s − 2.74·19-s − 1.41i·21-s + (4.71 + 0.896i)23-s + (0.900 − 4.91i)25-s + 2.52·27-s − 1.62·29-s − 4.55i·31-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + (0.768 − 0.640i)5-s + 0.242i·7-s + 0.618·9-s − 0.865·11-s + 0.112i·13-s + (−0.977 + 0.814i)15-s + 1.19·17-s − 0.629·19-s − 0.308i·21-s + (0.982 + 0.186i)23-s + (0.180 − 0.983i)25-s + 0.485·27-s − 0.302·29-s − 0.818i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9009745776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9009745776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.71 + 1.43i)T \) |
| 23 | \( 1 + (-4.71 - 0.896i)T \) |
| good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 7 | \( 1 - 0.642iT - 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 0.405iT - 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 + 4.55iT - 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 3.29iT - 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 - 0.263T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.19iT - 61T^{2} \) |
| 67 | \( 1 - 2.18iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 4.43iT - 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 + 9.14iT - 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.108037346646971134754830812777, −8.339646648644729327085062018051, −7.36387673170784030623067972798, −6.41182659657684970402907144795, −5.61629531948819068155888347525, −5.30068904535972875671493704317, −4.43113789434119370329119925216, −2.98291065552555635324298965064, −1.72586520542686239573870515712, −0.45024779792827326402977809464,
1.11874019054597947895387656079, 2.52374078305042023935495836050, 3.52799892585769011765832694894, 5.00732526544227170324398422703, 5.35506221892745118808548203796, 6.23879351124122981252025638506, 6.85243422754423856140049056091, 7.67739779971595627805123797421, 8.697707661332690698854252284559, 9.753555561887758970499919251095
