L(s) = 1 | + (0.602 + 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.260 + 0.673i)5-s + (−0.932 + 0.361i)8-s + (−0.0922 + 0.995i)9-s + (−0.694 + 0.197i)10-s + (−0.172 + 0.0666i)13-s + (−0.850 − 0.526i)16-s + (−0.850 − 0.526i)17-s + (−0.850 + 0.526i)18-s + (−0.576 − 0.435i)20-s + (0.353 + 0.322i)25-s + (−0.156 − 0.0971i)26-s + (1.91 + 0.544i)29-s + (−0.0922 − 0.995i)32-s + ⋯ |
L(s) = 1 | + (0.602 + 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.260 + 0.673i)5-s + (−0.932 + 0.361i)8-s + (−0.0922 + 0.995i)9-s + (−0.694 + 0.197i)10-s + (−0.172 + 0.0666i)13-s + (−0.850 − 0.526i)16-s + (−0.850 − 0.526i)17-s + (−0.850 + 0.526i)18-s + (−0.576 − 0.435i)20-s + (0.353 + 0.322i)25-s + (−0.156 − 0.0971i)26-s + (1.91 + 0.544i)29-s + (−0.0922 − 0.995i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.183986160\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183986160\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.602 - 0.798i)T \) |
| 17 | \( 1 + (0.850 + 0.526i)T \) |
good | 3 | \( 1 + (0.0922 - 0.995i)T^{2} \) |
| 5 | \( 1 + (0.260 - 0.673i)T + (-0.739 - 0.673i)T^{2} \) |
| 7 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 11 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 13 | \( 1 + (0.172 - 0.0666i)T + (0.739 - 0.673i)T^{2} \) |
| 19 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 23 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 29 | \( 1 + (-1.91 - 0.544i)T + (0.850 + 0.526i)T^{2} \) |
| 31 | \( 1 + (0.739 - 0.673i)T^{2} \) |
| 37 | \( 1 + (1.42 + 0.711i)T + (0.602 + 0.798i)T^{2} \) |
| 41 | \( 1 + (-0.486 - 0.533i)T + (-0.0922 + 0.995i)T^{2} \) |
| 43 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 47 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 53 | \( 1 + (0.181 - 1.95i)T + (-0.982 - 0.183i)T^{2} \) |
| 59 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 61 | \( 1 + (-0.247 + 1.32i)T + (-0.932 - 0.361i)T^{2} \) |
| 67 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 71 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 1.69i)T + (-0.445 - 0.895i)T^{2} \) |
| 79 | \( 1 + (-0.273 - 0.961i)T^{2} \) |
| 83 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 89 | \( 1 + (-1.12 - 0.435i)T + (0.739 + 0.673i)T^{2} \) |
| 97 | \( 1 + (-1.91 - 0.177i)T + (0.982 + 0.183i)T^{2} \) |
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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
−10.52810422704171881985710866720, −9.240705459823489903960490040524, −8.497132189606622578726294034372, −7.60640485177579600663603984755, −7.01236964699844598589966695910, −6.23429547882324734876050111544, −5.11418883936235938898482394623, −4.51159019299666831159097833799, −3.27893334233661753620396090532, −2.40379774017015247762345280619,
0.897721046801917854359794359452, 2.35903161048342478385520020906, 3.52917422790424494265918591619, 4.35832296196147792265836703596, 5.14362309299174823352268864773, 6.21966187630764150228683303417, 6.88733386295836778850570279689, 8.480901353470120633565860123785, 8.809894187112633317694290737392, 9.870698368113752987928581072023