| L(s) = 1 | + 0.381·2-s + 2.23·3-s − 1.85·4-s + 0.854·6-s − 0.236·7-s − 1.47·8-s + 2.00·9-s − 3.47·11-s − 4.14·12-s − 2·13-s − 0.0901·14-s + 3.14·16-s − 1.85·17-s + 0.763·18-s − 0.527·21-s − 1.32·22-s − 3.23·23-s − 3.29·24-s − 0.763·26-s − 2.23·27-s + 0.437·28-s − 6·29-s − 4.85·31-s + 4.14·32-s − 7.76·33-s − 0.708·34-s − 3.70·36-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 1.29·3-s − 0.927·4-s + 0.348·6-s − 0.0892·7-s − 0.520·8-s + 0.666·9-s − 1.04·11-s − 1.19·12-s − 0.554·13-s − 0.0240·14-s + 0.786·16-s − 0.449·17-s + 0.180·18-s − 0.115·21-s − 0.282·22-s − 0.674·23-s − 0.671·24-s − 0.149·26-s − 0.430·27-s + 0.0827·28-s − 1.11·29-s − 0.871·31-s + 0.732·32-s − 1.35·33-s − 0.121·34-s − 0.618·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007738542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007738542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 + 8.61T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−7.70946533386677580161801671705, −7.53199758408307633699694527820, −6.23808047807277525242572979983, −5.56275845217199680373185068814, −4.86032696566018611235211145461, −4.03933060691991415349575302132, −3.57101228384936875797749352760, −2.58267636702775333577416494771, −2.18123886133608093418840986860, −0.58815393720946770600808150623,
0.58815393720946770600808150623, 2.18123886133608093418840986860, 2.58267636702775333577416494771, 3.57101228384936875797749352760, 4.03933060691991415349575302132, 4.86032696566018611235211145461, 5.56275845217199680373185068814, 6.23808047807277525242572979983, 7.53199758408307633699694527820, 7.70946533386677580161801671705
