| L(s) = 1 | − 1.94·2-s + 3-s + 1.77·4-s + 3.66·5-s − 1.94·6-s + 7-s + 0.445·8-s + 9-s − 7.11·10-s + 2.76·11-s + 1.77·12-s − 3.90·13-s − 1.94·14-s + 3.66·15-s − 4.40·16-s + 3.65·17-s − 1.94·18-s − 2.93·19-s + 6.49·20-s + 21-s − 5.36·22-s − 5.90·23-s + 0.445·24-s + 8.44·25-s + 7.57·26-s + 27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.577·3-s + 0.885·4-s + 1.63·5-s − 0.792·6-s + 0.377·7-s + 0.157·8-s + 0.333·9-s − 2.25·10-s + 0.833·11-s + 0.511·12-s − 1.08·13-s − 0.518·14-s + 0.946·15-s − 1.10·16-s + 0.886·17-s − 0.457·18-s − 0.673·19-s + 1.45·20-s + 0.218·21-s − 1.14·22-s − 1.23·23-s + 0.0909·24-s + 1.68·25-s + 1.48·26-s + 0.192·27-s + 0.334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658333257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658333257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 0.885T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 4.06T + 71T^{2} \) |
| 73 | \( 1 - 1.01T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 - 3.97T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.146991980132209656042506457977, −8.185733634805425216259533516960, −7.64412600356226625902193621702, −6.72458533232702536920123632722, −6.01858720460623251053275636126, −5.02500502677177488550217051099, −4.04003608778331298892364885994, −2.49074495906586659618172567654, −1.98882658606298297176312518928, −1.02594787257228366011201918454,
1.02594787257228366011201918454, 1.98882658606298297176312518928, 2.49074495906586659618172567654, 4.04003608778331298892364885994, 5.02500502677177488550217051099, 6.01858720460623251053275636126, 6.72458533232702536920123632722, 7.64412600356226625902193621702, 8.185733634805425216259533516960, 9.146991980132209656042506457977
