| L(s) = 1 | + 1.77·2-s − 2.61·3-s + 1.14·4-s + 2.86·5-s − 4.64·6-s + 2.04·7-s − 1.52·8-s + 3.85·9-s + 5.07·10-s − 3.53·11-s − 2.99·12-s + 5.12·13-s + 3.63·14-s − 7.49·15-s − 4.97·16-s + 5.62·17-s + 6.83·18-s + 4.67·19-s + 3.26·20-s − 5.36·21-s − 6.26·22-s − 1.06·23-s + 3.98·24-s + 3.19·25-s + 9.08·26-s − 2.24·27-s + 2.34·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 1.51·3-s + 0.570·4-s + 1.28·5-s − 1.89·6-s + 0.774·7-s − 0.537·8-s + 1.28·9-s + 1.60·10-s − 1.06·11-s − 0.863·12-s + 1.42·13-s + 0.971·14-s − 1.93·15-s − 1.24·16-s + 1.36·17-s + 1.61·18-s + 1.07·19-s + 0.731·20-s − 1.17·21-s − 1.33·22-s − 0.221·23-s + 0.812·24-s + 0.639·25-s + 1.78·26-s − 0.432·27-s + 0.442·28-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
| \(L(1)\) |
\(\approx\) |
\(2.183429548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183429548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.77T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 8.34T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 1.19T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 6.59T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 8.64T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−10.76704717240101632079015379938, −10.20036581766084799150301775478, −9.025900766687343736359704296677, −7.69236383858799453328088604510, −6.37706266533298875489777900286, −5.60937980820098280442345897087, −5.47458109263617739811987622505, −4.47901046129733400605351991646, −3.01209693063152303368128706889, −1.32197642990595936093293754730,
1.32197642990595936093293754730, 3.01209693063152303368128706889, 4.47901046129733400605351991646, 5.47458109263617739811987622505, 5.60937980820098280442345897087, 6.37706266533298875489777900286, 7.69236383858799453328088604510, 9.025900766687343736359704296677, 10.20036581766084799150301775478, 10.76704717240101632079015379938
