L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s − 4·11-s + 2·14-s + 16-s − 4·17-s − 6·19-s + 20-s + 4·22-s + 25-s − 2·28-s − 4·29-s − 6·31-s − 32-s + 4·34-s − 2·35-s + 2·37-s + 6·38-s − 40-s − 10·41-s + 8·43-s − 4·44-s − 3·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.534·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.377·28-s − 0.742·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s − 0.338·35-s + 0.328·37-s + 0.973·38-s − 0.158·40-s − 1.56·41-s + 1.21·43-s − 0.603·44-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3792603609\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3792603609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−16.11437148571147, −15.56264485866982, −15.08507328198999, −14.52386917906796, −13.65751026139822, −13.04940092568306, −12.88475940459361, −12.18242824645059, −11.25522382619106, −10.82754532077745, −10.36018787830020, −9.782791931314537, −9.115006122141466, −8.731851361107938, −7.969680129119810, −7.387880056980364, −6.657558627613138, −6.203899510819719, −5.508390974040496, −4.763782462957253, −3.883283774806463, −3.020744515218506, −2.319418026039772, −1.723892495978389, −0.2795439725135611,
0.2795439725135611, 1.723892495978389, 2.319418026039772, 3.020744515218506, 3.883283774806463, 4.763782462957253, 5.508390974040496, 6.203899510819719, 6.657558627613138, 7.387880056980364, 7.969680129119810, 8.731851361107938, 9.115006122141466, 9.782791931314537, 10.36018787830020, 10.82754532077745, 11.25522382619106, 12.18242824645059, 12.88475940459361, 13.04940092568306, 13.65751026139822, 14.52386917906796, 15.08507328198999, 15.56264485866982, 16.11437148571147