| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s + 21-s + 22-s + 23-s − 24-s + 26-s − 27-s − 28-s − 29-s − 31-s + 32-s − 33-s + 34-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s + 21-s + 22-s + 23-s − 24-s + 26-s − 27-s − 28-s − 29-s − 31-s + 32-s − 33-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.602353557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602353557\) |
| \(L(1)\) |
\(\approx\) |
\(1.444758180\) |
| \(L(1)\) |
\(\approx\) |
\(1.444758180\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 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
−27.56261263634194716472854546570, −25.928354496995481467781444836371, −25.18000689054651910641174787632, −24.07080458235975675122735867091, −23.06566877175269978943550029361, −22.71381167016355623362571163988, −21.68734606842855070236525015718, −20.85949755955227770309136893657, −19.51822693391218852143517717408, −18.67202093479670079934300751515, −17.039538631552477881723233953698, −16.492612220777788609883998317542, −15.53911751700895026450488121430, −14.42193939858461021945401179129, −13.06263412067005051506116096806, −12.549448918656263324766606480221, −11.39771440935619170257200036759, −10.6112751447572385521452831792, −9.29848224275747034454853841715, −7.35321564189140194602436512335, −6.34465709351130316356931623066, −5.74139637456134224775425522400, −4.28958219905964262240681109960, −3.34972409903443562740326122793, −1.42759984804013631016017777792,
1.42759984804013631016017777792, 3.34972409903443562740326122793, 4.28958219905964262240681109960, 5.74139637456134224775425522400, 6.34465709351130316356931623066, 7.35321564189140194602436512335, 9.29848224275747034454853841715, 10.6112751447572385521452831792, 11.39771440935619170257200036759, 12.549448918656263324766606480221, 13.06263412067005051506116096806, 14.42193939858461021945401179129, 15.53911751700895026450488121430, 16.492612220777788609883998317542, 17.039538631552477881723233953698, 18.67202093479670079934300751515, 19.51822693391218852143517717408, 20.85949755955227770309136893657, 21.68734606842855070236525015718, 22.71381167016355623362571163988, 23.06566877175269978943550029361, 24.07080458235975675122735867091, 25.18000689054651910641174787632, 25.928354496995481467781444836371, 27.56261263634194716472854546570
