L(s) = 1 | + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (1.02 − 0.634i)5-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s − 1.20·10-s + (−0.876 + 0.163i)13-s + (−0.602 + 0.798i)16-s + (0.0170 + 0.183i)17-s + (−0.273 − 0.961i)18-s + (1.02 + 0.634i)20-s + (0.201 − 0.405i)25-s + (0.831 + 0.322i)26-s + (1.18 − 1.56i)29-s + (0.932 − 0.361i)32-s + ⋯ |
L(s) = 1 | + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (1.02 − 0.634i)5-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s − 1.20·10-s + (−0.876 + 0.163i)13-s + (−0.602 + 0.798i)16-s + (0.0170 + 0.183i)17-s + (−0.273 − 0.961i)18-s + (1.02 + 0.634i)20-s + (0.201 − 0.405i)25-s + (0.831 + 0.322i)26-s + (1.18 − 1.56i)29-s + (0.932 − 0.361i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7352308366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7352308366\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.850 + 0.526i)T \) |
| 137 | \( 1 + (0.982 + 0.183i)T \) |
good | 3 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 5 | \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \) |
| 7 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 11 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 13 | \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \) |
| 17 | \( 1 + (-0.0170 - 0.183i)T + (-0.982 + 0.183i)T^{2} \) |
| 19 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 23 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 29 | \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \) |
| 31 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 37 | \( 1 + 1.70T + T^{2} \) |
| 41 | \( 1 + 0.547T + T^{2} \) |
| 43 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 47 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 53 | \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \) |
| 59 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 61 | \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \) |
| 67 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 71 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 73 | \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \) |
| 79 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 83 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 89 | \( 1 + (1.58 - 0.981i)T + (0.445 - 0.895i)T^{2} \) |
| 97 | \( 1 + (-0.891 - 1.79i)T + (-0.602 + 0.798i)T^{2} \) |
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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
−10.51813349439605944122039818770, −10.07250032434451098147870220013, −9.332846137530892953969422826019, −8.454069611312507638214846407596, −7.52366693578752657067105239073, −6.57807231473082416729138127166, −5.27932026831383470651153221121, −4.21687931675048143212310111655, −2.53606125521311599423913339256, −1.56660473720975213498296493320,
1.64088918928215887831913089905, 2.95748019265929001719007852377, 4.82246872082922944008096649610, 5.85328483367670936756980709188, 6.80966951473213204639032514337, 7.24754791316507779029579761123, 8.597147424214292289997781631968, 9.403492638037084238136529352574, 10.20426696055356966327405039790, 10.53320300920943658508933113790