L(s) = 1 | − 3-s + 9-s + 4·11-s − 13-s − 8·17-s − 5·19-s − 4·23-s − 27-s + 9·29-s + 4·31-s − 4·33-s + 3·37-s + 39-s + 5·41-s + 6·43-s − 5·47-s − 7·49-s + 8·51-s + 5·53-s + 5·57-s + 6·59-s + 4·61-s − 3·67-s + 4·69-s − 7·71-s − 4·73-s + 79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.94·17-s − 1.14·19-s − 0.834·23-s − 0.192·27-s + 1.67·29-s + 0.718·31-s − 0.696·33-s + 0.493·37-s + 0.160·39-s + 0.780·41-s + 0.914·43-s − 0.729·47-s − 49-s + 1.12·51-s + 0.686·53-s + 0.662·57-s + 0.781·59-s + 0.512·61-s − 0.366·67-s + 0.481·69-s − 0.830·71-s − 0.468·73-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.22132126621718, −15.80540972297407, −15.26103192376788, −14.51449434922601, −14.21653594638895, −13.35091745847076, −12.98390854083362, −12.25014096909965, −11.79429555298883, −11.25689233224949, −10.73694531584088, −10.05955853981932, −9.540408747528466, −8.659317936190414, −8.517411603144042, −7.492130401646268, −6.737906775139208, −6.380192117491822, −5.940054647786845, −4.767634777910741, −4.413880286080627, −3.887623830327501, −2.682676730145878, −2.045114406583356, −1.048021729301554, 0,
1.048021729301554, 2.045114406583356, 2.682676730145878, 3.887623830327501, 4.413880286080627, 4.767634777910741, 5.940054647786845, 6.380192117491822, 6.737906775139208, 7.492130401646268, 8.517411603144042, 8.659317936190414, 9.540408747528466, 10.05955853981932, 10.73694531584088, 11.25689233224949, 11.79429555298883, 12.25014096909965, 12.98390854083362, 13.35091745847076, 14.21653594638895, 14.51449434922601, 15.26103192376788, 15.80540972297407, 16.22132126621718