| L(s) = 1 | + 0.381·2-s − 3-s − 1.85·4-s + 1.61·5-s − 0.381·6-s + 7-s − 1.47·8-s + 9-s + 0.618·10-s + 1.85·12-s + 4.23·13-s + 0.381·14-s − 1.61·15-s + 3.14·16-s + 7.85·17-s + 0.381·18-s + 0.854·19-s − 3·20-s − 21-s − 4.23·23-s + 1.47·24-s − 2.38·25-s + 1.61·26-s − 27-s − 1.85·28-s + 6·29-s − 0.618·30-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.723·5-s − 0.155·6-s + 0.377·7-s − 0.520·8-s + 0.333·9-s + 0.195·10-s + 0.535·12-s + 1.17·13-s + 0.102·14-s − 0.417·15-s + 0.786·16-s + 1.90·17-s + 0.0900·18-s + 0.195·19-s − 0.670·20-s − 0.218·21-s − 0.883·23-s + 0.300·24-s − 0.476·25-s + 0.317·26-s − 0.192·27-s − 0.350·28-s + 1.11·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283749078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283749078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−11.61215141032998682664216013847, −10.25337631368427615182969572735, −9.861919358650799954355187995432, −8.641764915832080958808547938776, −7.80965097349434811601625074813, −6.18219234634381625701084691188, −5.63369731786823936784536233854, −4.59053258519555098502675020313, −3.37718679860900972842092947176, −1.26179673917819932488302670889,
1.26179673917819932488302670889, 3.37718679860900972842092947176, 4.59053258519555098502675020313, 5.63369731786823936784536233854, 6.18219234634381625701084691188, 7.80965097349434811601625074813, 8.641764915832080958808547938776, 9.861919358650799954355187995432, 10.25337631368427615182969572735, 11.61215141032998682664216013847
