| L(s) = 1 | − 1.29·2-s + 3.40·3-s − 0.324·4-s + 2.94·5-s − 4.41·6-s − 0.962·7-s + 3.00·8-s + 8.61·9-s − 3.80·10-s − 11-s − 1.10·12-s − 1.22·13-s + 1.24·14-s + 10.0·15-s − 3.24·16-s + 4.62·17-s − 11.1·18-s − 5.94·19-s − 0.953·20-s − 3.27·21-s + 1.29·22-s − 0.745·23-s + 10.2·24-s + 3.65·25-s + 1.58·26-s + 19.1·27-s + 0.312·28-s + ⋯ |
| L(s) = 1 | − 0.915·2-s + 1.96·3-s − 0.162·4-s + 1.31·5-s − 1.80·6-s − 0.363·7-s + 1.06·8-s + 2.87·9-s − 1.20·10-s − 0.301·11-s − 0.318·12-s − 0.339·13-s + 0.333·14-s + 2.58·15-s − 0.811·16-s + 1.12·17-s − 2.62·18-s − 1.36·19-s − 0.213·20-s − 0.715·21-s + 0.275·22-s − 0.155·23-s + 2.09·24-s + 0.731·25-s + 0.311·26-s + 3.67·27-s + 0.0589·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011892584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011892584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 0.962T + 7T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 + 0.745T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 + 7.01T + 31T^{2} \) |
| 37 | \( 1 - 8.66T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 5.44T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−10.03577321767184995913525722796, −9.414825297941175488476085759109, −9.064076523530653221876484987650, −8.033554152583894565350464430738, −7.54206412400215578766980884647, −6.31916789862838907184871549348, −4.83154389702530355781251950120, −3.64449285955004886054966007929, −2.42056058734467106636810439923, −1.59494850362509287352611003543,
1.59494850362509287352611003543, 2.42056058734467106636810439923, 3.64449285955004886054966007929, 4.83154389702530355781251950120, 6.31916789862838907184871549348, 7.54206412400215578766980884647, 8.033554152583894565350464430738, 9.064076523530653221876484987650, 9.414825297941175488476085759109, 10.03577321767184995913525722796
