L(s) = 1 | − 1.26·2-s + 3.28·3-s − 0.388·4-s + 2.23·5-s − 4.17·6-s + 3.15·7-s + 3.03·8-s + 7.80·9-s − 2.83·10-s + 1.25·11-s − 1.27·12-s + 0.135·13-s − 4.00·14-s + 7.34·15-s − 3.07·16-s − 8.12·17-s − 9.90·18-s − 6.16·19-s − 0.869·20-s + 10.3·21-s − 1.59·22-s − 6.32·23-s + 9.96·24-s − 0.00254·25-s − 0.172·26-s + 15.7·27-s − 1.22·28-s + ⋯ |
L(s) = 1 | − 0.897·2-s + 1.89·3-s − 0.194·4-s + 0.999·5-s − 1.70·6-s + 1.19·7-s + 1.07·8-s + 2.60·9-s − 0.897·10-s + 0.378·11-s − 0.369·12-s + 0.0376·13-s − 1.06·14-s + 1.89·15-s − 0.767·16-s − 1.97·17-s − 2.33·18-s − 1.41·19-s − 0.194·20-s + 2.26·21-s − 0.339·22-s − 1.31·23-s + 2.03·24-s − 0.000509·25-s − 0.0337·26-s + 3.03·27-s − 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.057295930\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.057295930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 619 | \( 1 - T \) |
good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 - 3.28T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 - 0.135T + 13T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 0.790T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 3.99T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.94T + 89T^{2} \) |
| 97 | \( 1 + 1.33T + 97T^{2} \) |
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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
−10.31267122868205272446385112493, −9.362251649607263615945472676547, −8.876412216699152717861158198433, −8.334723114874212027495087634497, −7.57285870145561503082029340158, −6.47767927838726558072037285826, −4.68311997632134368178318646214, −4.02560500028598948480984417894, −2.10147256130631049724236624645, −1.85280751890730749370709431071,
1.85280751890730749370709431071, 2.10147256130631049724236624645, 4.02560500028598948480984417894, 4.68311997632134368178318646214, 6.47767927838726558072037285826, 7.57285870145561503082029340158, 8.334723114874212027495087634497, 8.876412216699152717861158198433, 9.362251649607263615945472676547, 10.31267122868205272446385112493