L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 15-s + 16-s + 6·17-s + 18-s − 20-s + 4·22-s − 8·23-s + 24-s + 25-s + 27-s + 10·29-s − 30-s − 8·31-s + 32-s + 4·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.285848739\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.285848739\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.72373262850118, −12.40222345160165, −11.96285854567205, −11.75173008722610, −11.14463247756419, −10.46478863264330, −10.11421327843777, −9.711678923858693, −9.060137347233257, −8.545321844404977, −8.229812259152852, −7.521379516186666, −7.261411022820034, −6.743347455855132, −5.980377675882019, −5.859768061174512, −5.069272004978112, −4.507014732864872, −3.953138097688228, −3.630259190961427, −3.235344116319972, −2.407390676567886, −1.981267276012844, −1.182069315377889, −0.6761045295338733,
0.6761045295338733, 1.182069315377889, 1.981267276012844, 2.407390676567886, 3.235344116319972, 3.630259190961427, 3.953138097688228, 4.507014732864872, 5.069272004978112, 5.859768061174512, 5.980377675882019, 6.743347455855132, 7.261411022820034, 7.521379516186666, 8.229812259152852, 8.545321844404977, 9.060137347233257, 9.711678923858693, 10.11421327843777, 10.46478863264330, 11.14463247756419, 11.75173008722610, 11.96285854567205, 12.40222345160165, 12.72373262850118