L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s + 2·12-s + 4·13-s + 14-s + 16-s − 6·17-s − 18-s − 2·19-s − 2·21-s − 2·24-s − 5·25-s − 4·26-s − 4·27-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s + 2·37-s + 2·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.436·21-s − 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.67574201633844, −15.46679125748719, −14.58701808015804, −14.14163717079093, −13.60488616815462, −13.12658681943917, −12.60823171198285, −11.81780269957263, −11.17973811716215, −10.77913807067584, −10.14536578893358, −9.391190451843792, −9.056556121843984, −8.576574175927213, −8.131298345473117, −7.468527917167834, −6.867618398365792, −6.143647793232479, −5.766722601415910, −4.550339944301472, −3.929088507428333, −3.354389623581556, −2.468691340059010, −2.143610866602282, −1.115511434245084, 0,
1.115511434245084, 2.143610866602282, 2.468691340059010, 3.354389623581556, 3.929088507428333, 4.550339944301472, 5.766722601415910, 6.143647793232479, 6.867618398365792, 7.468527917167834, 8.131298345473117, 8.576574175927213, 9.056556121843984, 9.391190451843792, 10.14536578893358, 10.77913807067584, 11.17973811716215, 11.81780269957263, 12.60823171198285, 13.12658681943917, 13.60488616815462, 14.14163717079093, 14.58701808015804, 15.46679125748719, 15.67574201633844