| L(s) = 1 | + (−0.420 − 0.902i)2-s + (1.48 − 0.890i)3-s + (0.648 − 0.772i)4-s + (−1.24 − 1.77i)5-s + (−1.42 − 0.965i)6-s + (−0.867 + 4.92i)7-s + (−2.89 − 0.775i)8-s + (1.41 − 2.64i)9-s + (−1.08 + 1.87i)10-s + (2.36 + 4.09i)11-s + (0.275 − 1.72i)12-s + (−0.286 − 3.27i)13-s + (4.80 − 1.28i)14-s + (−3.43 − 1.53i)15-s + (0.167 + 0.950i)16-s + (0.0360 − 0.412i)17-s + ⋯ |
| L(s) = 1 | + (−0.297 − 0.638i)2-s + (0.857 − 0.514i)3-s + (0.324 − 0.386i)4-s + (−0.557 − 0.795i)5-s + (−0.583 − 0.394i)6-s + (−0.328 + 1.86i)7-s + (−1.02 − 0.274i)8-s + (0.471 − 0.881i)9-s + (−0.341 + 0.592i)10-s + (0.712 + 1.23i)11-s + (0.0794 − 0.498i)12-s + (−0.0794 − 0.907i)13-s + (1.28 − 0.344i)14-s + (−0.886 − 0.396i)15-s + (0.0419 + 0.237i)16-s + (0.00874 − 0.0999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.828111 - 0.756861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.828111 - 0.756861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.48 + 0.890i)T \) |
| 37 | \( 1 + (3.19 - 5.17i)T \) |
| good | 2 | \( 1 + (0.420 + 0.902i)T + (-1.28 + 1.53i)T^{2} \) |
| 5 | \( 1 + (1.24 + 1.77i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (0.867 - 4.92i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.286 + 3.27i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.0360 + 0.412i)T + (-16.7 - 2.95i)T^{2} \) |
| 19 | \( 1 + (-1.79 - 0.837i)T + (12.2 + 14.5i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 3.76i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.994 - 3.71i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (0.758 - 0.758i)T - 31iT^{2} \) |
| 41 | \( 1 + (-0.196 - 0.165i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.12 + 4.12i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.62 + 4.40i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.241i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.237 - 0.166i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-9.17 + 0.802i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-0.0440 - 0.00775i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.72 + 7.47i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 0.498iT - 73T^{2} \) |
| 79 | \( 1 + (5.22 - 3.65i)T + (27.0 - 74.2i)T^{2} \) |
| 83 | \( 1 + (-5.58 - 6.65i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.80 + 12.5i)T + (-30.4 - 83.6i)T^{2} \) |
| 97 | \( 1 + (8.93 - 2.39i)T + (84.0 - 48.5i)T^{2} \) |
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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
−12.95404718317913314192718524840, −12.18262990758958245540439145243, −11.82931129887778107065542031017, −9.880576599924526065852570851774, −9.165651576050322875140357645542, −8.312381572997321280887711169451, −6.79308806575048838080803985967, −5.33817259359320987009462529425, −3.18056392625814234300434230361, −1.79453031087762642762153832587,
3.24059252130740229351439293072, 4.03525094835062808694257117186, 6.61353015937103281948323117427, 7.30193971154940650988953524338, 8.268697827618198678022459904252, 9.465867514412131919200496684710, 10.76530789923785301380924366622, 11.46197390470465465961581070206, 13.25737817200982564292412066241, 14.19406850198559141086805136720
