| L(s) = 1 | − 1.51·2-s + 3.11·3-s + 0.307·4-s + 1.87·5-s − 4.73·6-s − 0.343·7-s + 2.57·8-s + 6.70·9-s − 2.84·10-s + 1.29·11-s + 0.959·12-s + 1.43·13-s + 0.521·14-s + 5.83·15-s − 4.52·16-s − 1.26·17-s − 10.1·18-s + 4.20·19-s + 0.577·20-s − 1.06·21-s − 1.96·22-s − 8.00·23-s + 8.00·24-s − 1.48·25-s − 2.18·26-s + 11.5·27-s − 0.105·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 1.79·3-s + 0.153·4-s + 0.838·5-s − 1.93·6-s − 0.129·7-s + 0.908·8-s + 2.23·9-s − 0.900·10-s + 0.390·11-s + 0.276·12-s + 0.398·13-s + 0.139·14-s + 1.50·15-s − 1.13·16-s − 0.306·17-s − 2.39·18-s + 0.965·19-s + 0.129·20-s − 0.233·21-s − 0.419·22-s − 1.66·23-s + 1.63·24-s − 0.297·25-s − 0.427·26-s + 2.21·27-s − 0.0199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.878660339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878660339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 + 0.343T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 + 8.00T + 23T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 4.56T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 0.692T + 89T^{2} \) |
| 97 | \( 1 + 9.50T + 97T^{2} \) |
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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
−9.732139526581757397543010328083, −9.443368331110875900934977161728, −8.564615302518252961508848549979, −8.024470337397649253677519139791, −7.22980768462754860133756575248, −6.11054942057808531214574353752, −4.54364122946016928439572769277, −3.55489400144614801589377661673, −2.30332720740516342424368601028, −1.43412250235075595051148600127,
1.43412250235075595051148600127, 2.30332720740516342424368601028, 3.55489400144614801589377661673, 4.54364122946016928439572769277, 6.11054942057808531214574353752, 7.22980768462754860133756575248, 8.024470337397649253677519139791, 8.564615302518252961508848549979, 9.443368331110875900934977161728, 9.732139526581757397543010328083
