| L(s) = 1 | + 3-s − 5-s − 2·9-s − 6·11-s + 2·13-s − 15-s + 6·17-s + 8·19-s + 3·23-s + 25-s − 5·27-s − 3·29-s − 2·31-s − 6·33-s − 8·37-s + 2·39-s + 3·41-s − 5·43-s + 2·45-s + 6·51-s − 12·53-s + 6·55-s + 8·57-s − 61-s − 2·65-s + 7·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.359·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s + 0.468·41-s − 0.762·43-s + 0.298·45-s + 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s − 0.128·61-s − 0.248·65-s + 0.855·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816795573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816795573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−15.90889178326795, −15.52290697580212, −14.89259234832691, −14.23605694963095, −13.88766712368576, −13.25339642521316, −12.73830650796264, −12.07132543621424, −11.49492200705564, −10.91719268016665, −10.37797362981413, −9.645890656819274, −9.186551340721934, −8.346688800670895, −7.878472316419055, −7.610896209518821, −6.839377679207846, −5.718304933565735, −5.392083740367081, −4.865763029819278, −3.524794229377930, −3.316446484380378, −2.690608758997877, −1.646000676638222, −0.5605304428189968,
0.5605304428189968, 1.646000676638222, 2.690608758997877, 3.316446484380378, 3.524794229377930, 4.865763029819278, 5.392083740367081, 5.718304933565735, 6.839377679207846, 7.610896209518821, 7.878472316419055, 8.346688800670895, 9.186551340721934, 9.645890656819274, 10.37797362981413, 10.91719268016665, 11.49492200705564, 12.07132543621424, 12.73830650796264, 13.25339642521316, 13.88766712368576, 14.23605694963095, 14.89259234832691, 15.52290697580212, 15.90889178326795
