| L(s) = 1 | − 5-s + 2·7-s − 2·11-s + 6·13-s + 4·17-s − 4·19-s + 25-s − 2·29-s − 4·31-s − 2·35-s − 8·37-s − 8·41-s − 43-s − 8·47-s − 3·49-s + 2·53-s + 2·55-s + 6·59-s + 4·61-s − 6·65-s + 12·67-s + 10·73-s − 4·77-s + 8·79-s − 14·83-s − 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.603·11-s + 1.66·13-s + 0.970·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.338·35-s − 1.31·37-s − 1.24·41-s − 0.152·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.269·55-s + 0.781·59-s + 0.512·61-s − 0.744·65-s + 1.46·67-s + 1.17·73-s − 0.455·77-s + 0.900·79-s − 1.53·83-s − 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.75359741724850, −13.28613945144120, −12.80115186942224, −12.39807576042523, −11.73739774737780, −11.30428997441058, −10.93810702075941, −10.52780131561772, −9.949656017242314, −9.413873507520246, −8.576394696861264, −8.264470898914106, −8.213524993177391, −7.335325311685135, −6.860379157684104, −6.338458820721881, −5.584674560639367, −5.290122713552335, −4.696345691643436, −3.926430556121477, −3.571632395689173, −3.066034181571860, −2.040953934869862, −1.654688762874869, −0.8881919415709674, 0,
0.8881919415709674, 1.654688762874869, 2.040953934869862, 3.066034181571860, 3.571632395689173, 3.926430556121477, 4.696345691643436, 5.290122713552335, 5.584674560639367, 6.338458820721881, 6.860379157684104, 7.335325311685135, 8.213524993177391, 8.264470898914106, 8.576394696861264, 9.413873507520246, 9.949656017242314, 10.52780131561772, 10.93810702075941, 11.30428997441058, 11.73739774737780, 12.39807576042523, 12.80115186942224, 13.28613945144120, 13.75359741724850
