L(s) = 1 | + (0.526 − 0.850i)2-s + (−0.445 − 0.895i)4-s + (0.982 − 1.18i)5-s + (−0.995 − 0.0922i)8-s + (−0.361 + 0.932i)9-s + (−0.488 − 1.45i)10-s + (0.0666 − 0.719i)13-s + (−0.602 + 0.798i)16-s + (0.602 − 0.798i)17-s + (0.602 + 0.798i)18-s + (−1.49 − 0.352i)20-s + (−0.251 − 1.34i)25-s + (−0.576 − 0.435i)26-s + (−0.581 + 1.73i)29-s + (0.361 + 0.932i)32-s + ⋯ |
L(s) = 1 | + (0.526 − 0.850i)2-s + (−0.445 − 0.895i)4-s + (0.982 − 1.18i)5-s + (−0.995 − 0.0922i)8-s + (−0.361 + 0.932i)9-s + (−0.488 − 1.45i)10-s + (0.0666 − 0.719i)13-s + (−0.602 + 0.798i)16-s + (0.602 − 0.798i)17-s + (0.602 + 0.798i)18-s + (−1.49 − 0.352i)20-s + (−0.251 − 1.34i)25-s + (−0.576 − 0.435i)26-s + (−0.581 + 1.73i)29-s + (0.361 + 0.932i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.440863887\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.440863887\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.526 + 0.850i)T \) |
| 17 | \( 1 + (-0.602 + 0.798i)T \) |
good | 3 | \( 1 + (0.361 - 0.932i)T^{2} \) |
| 5 | \( 1 + (-0.982 + 1.18i)T + (-0.183 - 0.982i)T^{2} \) |
| 7 | \( 1 + (-0.673 + 0.739i)T^{2} \) |
| 11 | \( 1 + (-0.798 - 0.602i)T^{2} \) |
| 13 | \( 1 + (-0.0666 + 0.719i)T + (-0.982 - 0.183i)T^{2} \) |
| 19 | \( 1 + (0.445 - 0.895i)T^{2} \) |
| 23 | \( 1 + (-0.673 + 0.739i)T^{2} \) |
| 29 | \( 1 + (0.581 - 1.73i)T + (-0.798 - 0.602i)T^{2} \) |
| 31 | \( 1 + (-0.183 + 0.982i)T^{2} \) |
| 37 | \( 1 + (1.70 - 0.947i)T + (0.526 - 0.850i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 0.722i)T + (0.361 - 0.932i)T^{2} \) |
| 43 | \( 1 + (-0.273 + 0.961i)T^{2} \) |
| 47 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 53 | \( 1 + (0.486 - 1.25i)T + (-0.739 - 0.673i)T^{2} \) |
| 59 | \( 1 + (0.0922 - 0.995i)T^{2} \) |
| 61 | \( 1 + (-1.64 + 0.0762i)T + (0.995 - 0.0922i)T^{2} \) |
| 67 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 71 | \( 1 + (0.673 - 0.739i)T^{2} \) |
| 73 | \( 1 + (0.800 - 0.111i)T + (0.961 - 0.273i)T^{2} \) |
| 79 | \( 1 + (0.895 + 0.445i)T^{2} \) |
| 83 | \( 1 + (0.932 - 0.361i)T^{2} \) |
| 89 | \( 1 + (-0.0971 - 1.04i)T + (-0.982 + 0.183i)T^{2} \) |
| 97 | \( 1 + (0.765 + 1.73i)T + (-0.673 + 0.739i)T^{2} \) |
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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
−9.896490513618054923832820375327, −8.992781653439499241162647361684, −8.491853356522543088448658587593, −7.21674057894720337812879834755, −5.79975697393362416254235905403, −5.30844483381953853917919128055, −4.77234444345632945432865716309, −3.37660097642310406507842978312, −2.28053024422713153202601316605, −1.22589710632426478956440652909,
2.21412855389388690522458511680, 3.31972999709667839033872111224, 4.11184494764402618009058260278, 5.57177199574147810716045109470, 6.10329128058874121551779158511, 6.71949215788156549927949681624, 7.51801635897850325534016108338, 8.574090953704376417196168085881, 9.426835984947397577002788627525, 10.04039608320705188061782303285