| L(s) = 1 | − 2.49·2-s + 3-s + 4.24·4-s − 2.35·5-s − 2.49·6-s + 7-s − 5.59·8-s + 9-s + 5.87·10-s − 5.16·11-s + 4.24·12-s − 2.49·14-s − 2.35·15-s + 5.50·16-s − 1.70·17-s − 2.49·18-s + 8.02·19-s − 9.97·20-s + 21-s + 12.9·22-s + 4.21·23-s − 5.59·24-s + 0.531·25-s + 27-s + 4.24·28-s − 5.24·29-s + 5.87·30-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.12·4-s − 1.05·5-s − 1.01·6-s + 0.377·7-s − 1.97·8-s + 0.333·9-s + 1.85·10-s − 1.55·11-s + 1.22·12-s − 0.667·14-s − 0.607·15-s + 1.37·16-s − 0.413·17-s − 0.588·18-s + 1.84·19-s − 2.23·20-s + 0.218·21-s + 2.75·22-s + 0.879·23-s − 1.14·24-s + 0.106·25-s + 0.192·27-s + 0.801·28-s − 0.973·29-s + 1.07·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5965104540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5965104540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 8.02T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + 0.965T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.318T + 71T^{2} \) |
| 73 | \( 1 - 0.542T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 - 0.287T + 97T^{2} \) |
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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
−8.500760697424152615451989509642, −7.905940653950803771157482920122, −7.39904086686861617624883173744, −7.08198583414414417971551240441, −5.65301943356881891864937021728, −4.79562991215742310516975418293, −3.50611866252528318224054355428, −2.77472433182063401000424405246, −1.77919396432603478799664946232, −0.56720486406037879253334173467,
0.56720486406037879253334173467, 1.77919396432603478799664946232, 2.77472433182063401000424405246, 3.50611866252528318224054355428, 4.79562991215742310516975418293, 5.65301943356881891864937021728, 7.08198583414414417971551240441, 7.39904086686861617624883173744, 7.905940653950803771157482920122, 8.500760697424152615451989509642
