| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.447952862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447952862\) |
| \(L(1)\) |
\(\approx\) |
\(1.464441402\) |
| \(L(1)\) |
\(\approx\) |
\(1.464441402\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 89 | \( 1 \) |
| 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 | \( 1 - T \) |
| 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
−29.98598621102226172172997306133, −29.73708082198394752880679181430, −28.79856429691325438555086791031, −27.73524911083589315304005394604, −25.965824155371753959884170982564, −25.03505813784555786923448783818, −24.04303523215240453311801876630, −22.846533689004537974296947389595, −22.08232475325210703896522981363, −21.49958406900697463175170672703, −20.00192495619448224409022514668, −18.71881712022508977228723702847, −16.94638287028052911981423016286, −16.724229938702739733300679890196, −15.13958911329370947924774246622, −13.92581184088056377045758907191, −12.71262220728461126154191301074, −12.05534923353645711137815985397, −10.52520691356189272358208959507, −9.63054086133726650443834762504, −7.060278444381633573094194937, −6.19688904057110273407365751639, −5.27470947518760387294831369951, −3.749232132316333766876802799234, −1.90110524970185567177794590654,
1.90110524970185567177794590654, 3.749232132316333766876802799234, 5.27470947518760387294831369951, 6.19688904057110273407365751639, 7.060278444381633573094194937, 9.63054086133726650443834762504, 10.52520691356189272358208959507, 12.05534923353645711137815985397, 12.71262220728461126154191301074, 13.92581184088056377045758907191, 15.13958911329370947924774246622, 16.724229938702739733300679890196, 16.94638287028052911981423016286, 18.71881712022508977228723702847, 20.00192495619448224409022514668, 21.49958406900697463175170672703, 22.08232475325210703896522981363, 22.846533689004537974296947389595, 24.04303523215240453311801876630, 25.03505813784555786923448783818, 25.965824155371753959884170982564, 27.73524911083589315304005394604, 28.79856429691325438555086791031, 29.73708082198394752880679181430, 29.98598621102226172172997306133
