| L(s) = 1 | + 1.61·2-s − 1.61·3-s + 0.618·4-s − 2.85·5-s − 2.61·6-s + 2.23·7-s − 2.23·8-s − 0.381·9-s − 4.61·10-s + 3.61·11-s − 1.00·12-s + 4.23·13-s + 3.61·14-s + 4.61·15-s − 4.85·16-s + 6.61·17-s − 0.618·18-s − 1.85·19-s − 1.76·20-s − 3.61·21-s + 5.85·22-s + 3.23·23-s + 3.61·24-s + 3.14·25-s + 6.85·26-s + 5.47·27-s + 1.38·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.934·3-s + 0.309·4-s − 1.27·5-s − 1.06·6-s + 0.845·7-s − 0.790·8-s − 0.127·9-s − 1.46·10-s + 1.09·11-s − 0.288·12-s + 1.17·13-s + 0.966·14-s + 1.19·15-s − 1.21·16-s + 1.60·17-s − 0.145·18-s − 0.425·19-s − 0.394·20-s − 0.789·21-s + 1.24·22-s + 0.674·23-s + 0.738·24-s + 0.629·25-s + 1.34·26-s + 1.05·27-s + 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664686632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664686632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 0.291T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 - 8.70T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−10.78977792006605257639889524606, −9.236513758723421639959699559730, −8.400171141400084472062697779541, −7.53113409041226249517258001987, −6.31995150566450563052947508536, −5.73295323938505671118471030771, −4.73007397926055604656404000282, −4.03329640757439290099282440047, −3.21220399004379989912457639689, −0.975820601596906827136295852782,
0.975820601596906827136295852782, 3.21220399004379989912457639689, 4.03329640757439290099282440047, 4.73007397926055604656404000282, 5.73295323938505671118471030771, 6.31995150566450563052947508536, 7.53113409041226249517258001987, 8.400171141400084472062697779541, 9.236513758723421639959699559730, 10.78977792006605257639889524606
