| L(s) = 1 | + 1.53·2-s + 1.64·3-s + 0.356·4-s − 4.31·5-s + 2.52·6-s − 5.09·7-s − 2.52·8-s − 0.285·9-s − 6.61·10-s + 3.05·11-s + 0.587·12-s + 1.83·13-s − 7.82·14-s − 7.10·15-s − 4.58·16-s + 1.91·17-s − 0.438·18-s + 0.990·19-s − 1.53·20-s − 8.39·21-s + 4.69·22-s + 1.50·23-s − 4.15·24-s + 13.5·25-s + 2.81·26-s − 5.41·27-s − 1.81·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.951·3-s + 0.178·4-s − 1.92·5-s + 1.03·6-s − 1.92·7-s − 0.891·8-s − 0.0951·9-s − 2.09·10-s + 0.921·11-s + 0.169·12-s + 0.508·13-s − 2.09·14-s − 1.83·15-s − 1.14·16-s + 0.465·17-s − 0.103·18-s + 0.227·19-s − 0.343·20-s − 1.83·21-s + 1.00·22-s + 0.313·23-s − 0.848·24-s + 2.71·25-s + 0.551·26-s − 1.04·27-s − 0.343·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + 4.31T + 5T^{2} \) |
| 7 | \( 1 + 5.09T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 0.990T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 + 9.90T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 0.290T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 + 0.908T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−10.11086448126997088739708043238, −8.901906727329291200457416491174, −8.735692965682764199869202418384, −7.33941978620883193340377075285, −6.66669861763576477363354022288, −5.46709358446462274807761161336, −3.88800225340117395366920044796, −3.62571979784969242470944891888, −3.05858782345958765141610530550, 0,
3.05858782345958765141610530550, 3.62571979784969242470944891888, 3.88800225340117395366920044796, 5.46709358446462274807761161336, 6.66669861763576477363354022288, 7.33941978620883193340377075285, 8.735692965682764199869202418384, 8.901906727329291200457416491174, 10.11086448126997088739708043238
