| L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s − 0.381·5-s + 0.618·6-s + 7-s + 2.23·8-s + 9-s + 0.236·10-s + 2.23·11-s + 1.61·12-s − 0.145·13-s − 0.618·14-s + 0.381·15-s + 1.85·16-s − 5.47·17-s − 0.618·18-s − 8.23·19-s + 0.618·20-s − 21-s − 1.38·22-s + 23-s − 2.23·24-s − 4.85·25-s + 0.0901·26-s − 27-s − 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.170·5-s + 0.252·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.0746·10-s + 0.674·11-s + 0.467·12-s − 0.0404·13-s − 0.165·14-s + 0.0986·15-s + 0.463·16-s − 1.32·17-s − 0.145·18-s − 1.88·19-s + 0.138·20-s − 0.218·21-s − 0.294·22-s + 0.208·23-s − 0.456·24-s − 0.970·25-s + 0.0176·26-s − 0.192·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.145T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 + 6.85T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−10.70132206775849479650400765813, −9.505786204436346415991662290088, −8.847186422737230578667744672458, −7.968331840466953306972192152846, −6.87542569106972419037511880281, −5.84992407040168717391639404792, −4.59403386190936932432158710194, −4.00687879834158679166073984332, −1.82813868592499159551825674143, 0,
1.82813868592499159551825674143, 4.00687879834158679166073984332, 4.59403386190936932432158710194, 5.84992407040168717391639404792, 6.87542569106972419037511880281, 7.968331840466953306972192152846, 8.847186422737230578667744672458, 9.505786204436346415991662290088, 10.70132206775849479650400765813
