| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1609 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1609 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.373452271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.373452271\) |
| \(L(1)\) |
\(\approx\) |
\(3.061169634\) |
| \(L(1)\) |
\(\approx\) |
\(3.061169634\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1609 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.609438977965708218771214613841, −19.81276836665437625176257697843, −19.31417578855128012589641539332, −18.40740110978783857870371626722, −17.34351198249547162026759829085, −16.503425999280140287143089936676, −15.76223026965370833372586478727, −15.10717518089408192025698751529, −14.19602715676399508887431623259, −13.77108239190727136437545207723, −13.03057004456575617562264306271, −12.65784727027169559627700235376, −11.46440194384662493168674139941, −10.49735511567523847212917680571, −9.79179356723587508039390415861, −9.0013643837918238056771087398, −8.233988966406724346101447885651, −6.84117457507107153962315080088, −6.518496376202876873623110504334, −5.79768311442578100691711600895, −4.46051103570323942759477784676, −3.82565157667150926892654640762, −3.03458473151681647415218252350, −2.12021521747340844177830581406, −1.47714168874092913484817753657,
1.47714168874092913484817753657, 2.12021521747340844177830581406, 3.03458473151681647415218252350, 3.82565157667150926892654640762, 4.46051103570323942759477784676, 5.79768311442578100691711600895, 6.518496376202876873623110504334, 6.84117457507107153962315080088, 8.233988966406724346101447885651, 9.0013643837918238056771087398, 9.79179356723587508039390415861, 10.49735511567523847212917680571, 11.46440194384662493168674139941, 12.65784727027169559627700235376, 13.03057004456575617562264306271, 13.77108239190727136437545207723, 14.19602715676399508887431623259, 15.10717518089408192025698751529, 15.76223026965370833372586478727, 16.503425999280140287143089936676, 17.34351198249547162026759829085, 18.40740110978783857870371626722, 19.31417578855128012589641539332, 19.81276836665437625176257697843, 20.609438977965708218771214613841
