| L(s) = 1 | + 2.17·2-s + 2.70·4-s − 5-s + 2.87·7-s + 1.53·8-s − 2.17·10-s + 4.24·11-s − 3.61·13-s + 6.24·14-s − 2.07·16-s + 6.63·17-s + 19-s − 2.70·20-s + 9.21·22-s + 4.63·23-s + 25-s − 7.85·26-s + 7.80·28-s − 0.986·29-s + 2.44·31-s − 7.58·32-s + 14.3·34-s − 2.87·35-s − 9.80·37-s + 2.17·38-s − 1.53·40-s − 6.92·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s − 0.447·5-s + 1.08·7-s + 0.544·8-s − 0.686·10-s + 1.28·11-s − 1.00·13-s + 1.66·14-s − 0.519·16-s + 1.60·17-s + 0.229·19-s − 0.605·20-s + 1.96·22-s + 0.965·23-s + 0.200·25-s − 1.53·26-s + 1.47·28-s − 0.183·29-s + 0.439·31-s − 1.34·32-s + 2.46·34-s − 0.486·35-s − 1.61·37-s + 0.352·38-s − 0.243·40-s − 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.765594013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.765594013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 + 0.986T + 29T^{2} \) |
| 31 | \( 1 - 2.44T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 0.921T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 + 0.340T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−10.43971001735791934994682964517, −9.344868666628155920776279366114, −8.325930662864200164803640516049, −7.33437819599551275271351986480, −6.62934035756182628231282134022, −5.34958582783449550924469938335, −4.91655533332516728784095642649, −3.89489126392538337415760344234, −3.07390976964066397645761340768, −1.56287838856803859113291587061,
1.56287838856803859113291587061, 3.07390976964066397645761340768, 3.89489126392538337415760344234, 4.91655533332516728784095642649, 5.34958582783449550924469938335, 6.62934035756182628231282134022, 7.33437819599551275271351986480, 8.325930662864200164803640516049, 9.344868666628155920776279366114, 10.43971001735791934994682964517
