L(s) = 1 | − 1.61·2-s + 1.61·4-s + 0.618·5-s − 8-s + 9-s − 1.00·10-s − 1.61·11-s − 1.61·13-s − 1.61·18-s + 0.618·19-s + 1.00·20-s + 2.61·22-s + 0.618·23-s − 0.618·25-s + 2.61·26-s − 1.61·31-s + 32-s + 1.61·36-s − 1.00·38-s − 0.618·40-s − 2.61·44-s + 0.618·45-s − 1.00·46-s + 49-s + 0.999·50-s − 2.61·52-s − 1.00·55-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 1.61·4-s + 0.618·5-s − 8-s + 9-s − 1.00·10-s − 1.61·11-s − 1.61·13-s − 1.61·18-s + 0.618·19-s + 1.00·20-s + 2.61·22-s + 0.618·23-s − 0.618·25-s + 2.61·26-s − 1.61·31-s + 32-s + 1.61·36-s − 1.00·38-s − 0.618·40-s − 2.61·44-s + 0.618·45-s − 1.00·46-s + 49-s + 0.999·50-s − 2.61·52-s − 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2888166339\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2888166339\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−15.09438089440591022273049327848, −13.49476825961692915439080389739, −12.41280998218477785723439677217, −10.84468432527915440022069005515, −10.00582281033173329485446747233, −9.369771806913826376759339060543, −7.80057654256192655708865026236, −7.13654942825154090841373972766, −5.19349650437579468072287387020, −2.25042492444233676394569070777,
2.25042492444233676394569070777, 5.19349650437579468072287387020, 7.13654942825154090841373972766, 7.80057654256192655708865026236, 9.369771806913826376759339060543, 10.00582281033173329485446747233, 10.84468432527915440022069005515, 12.41280998218477785723439677217, 13.49476825961692915439080389739, 15.09438089440591022273049327848