| L(s) = 1 | + (0.163 + 0.118i)2-s + (0.259 + 0.625i)3-s + (−0.605 − 1.86i)4-s + (−1.89 − 1.18i)5-s + (−0.0319 + 0.133i)6-s + (1.01 − 1.65i)7-s + (0.247 − 0.760i)8-s + (1.79 − 1.79i)9-s + (−0.168 − 0.418i)10-s + (−1.50 + 1.75i)11-s + (1.00 − 0.861i)12-s + (1.19 − 4.95i)13-s + (0.361 − 0.149i)14-s + (0.252 − 1.49i)15-s + (−3.03 + 2.20i)16-s + (3.09 + 2.64i)17-s + ⋯ |
| L(s) = 1 | + (0.115 + 0.0839i)2-s + (0.149 + 0.361i)3-s + (−0.302 − 0.931i)4-s + (−0.847 − 0.531i)5-s + (−0.0130 + 0.0542i)6-s + (0.383 − 0.625i)7-s + (0.0873 − 0.268i)8-s + (0.598 − 0.598i)9-s + (−0.0532 − 0.132i)10-s + (−0.453 + 0.530i)11-s + (0.291 − 0.248i)12-s + (0.330 − 1.37i)13-s + (0.0967 − 0.0400i)14-s + (0.0651 − 0.385i)15-s + (−0.759 + 0.552i)16-s + (0.751 + 0.641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.922512 - 0.625612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.922512 - 0.625612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.89 + 1.18i)T \) |
| 41 | \( 1 + (-6.31 + 1.06i)T \) |
| good | 2 | \( 1 + (-0.163 - 0.118i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.259 - 0.625i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-1.01 + 1.65i)T + (-3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (1.50 - 1.75i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 4.95i)T + (-11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (-3.09 - 2.64i)T + (2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.511 - 0.833i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (0.289 + 1.82i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.461 - 5.86i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.79 + 0.907i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.49 + 6.85i)T + (-21.7 - 29.9i)T^{2} \) |
| 43 | \( 1 + (3.48 - 4.79i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-0.308 + 0.188i)T + (21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (1.75 + 2.05i)T + (-8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (3.47 - 4.78i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.00 - 6.32i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-12.2 - 0.965i)T + (66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-9.91 - 8.46i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + 2.73iT - 73T^{2} \) |
| 79 | \( 1 + (-3.81 + 9.20i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.16 + 2.16i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.29 - 0.550i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-0.978 + 12.4i)T + (-95.8 - 15.1i)T^{2} \) |
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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
−12.54737708079417850345306001848, −10.94317950545679173280986461309, −10.36370097559258592108387577237, −9.380829309922822368725793667686, −8.207951242718159474634027248388, −7.25153416774095854827836300181, −5.71116941456698639995706113649, −4.64037342792982456155142830629, −3.70153985320022584996380810913, −1.03189559298775416461184104542,
2.43441050686139820898815719937, 3.78211920704479362110150664437, 4.93362278456644839557365511753, 6.71672559266973087411605886247, 7.74441172932632343813423300158, 8.286033066611289023692710170493, 9.496884152022377594203201383553, 11.05769703710888843314375407802, 11.68925468530103413332915124514, 12.47771166762085408396892637180
