L(s) = 1 | − 2.15·2-s + 2.64·4-s − 3.62·5-s − 2.27·7-s − 1.39·8-s + 7.81·10-s − 11-s + 1.23·13-s + 4.89·14-s − 2.28·16-s − 5.69·17-s − 2.89·19-s − 9.59·20-s + 2.15·22-s − 5.91·23-s + 8.13·25-s − 2.66·26-s − 6.00·28-s + 3.50·29-s − 2.50·31-s + 7.72·32-s + 12.2·34-s + 8.22·35-s − 0.333·37-s + 6.23·38-s + 5.05·40-s + 9.96·41-s + ⋯ |
L(s) = 1 | − 1.52·2-s + 1.32·4-s − 1.62·5-s − 0.858·7-s − 0.492·8-s + 2.47·10-s − 0.301·11-s + 0.343·13-s + 1.30·14-s − 0.572·16-s − 1.38·17-s − 0.664·19-s − 2.14·20-s + 0.459·22-s − 1.23·23-s + 1.62·25-s − 0.523·26-s − 1.13·28-s + 0.651·29-s − 0.450·31-s + 1.36·32-s + 2.10·34-s + 1.39·35-s − 0.0549·37-s + 1.01·38-s + 0.798·40-s + 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2304726193\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2304726193\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 7.15T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 + 2.91T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−10.03712980714911648556009677479, −9.145952634679239803781135370549, −8.410578009913751117388162404293, −7.88028994378907269348673165738, −6.99515174042409879718721289538, −6.30935408181637199032909397485, −4.54437380029603670247969439808, −3.68461309018722344999261665210, −2.31565266577776637577914817395, −0.45448446258251434250283040992,
0.45448446258251434250283040992, 2.31565266577776637577914817395, 3.68461309018722344999261665210, 4.54437380029603670247969439808, 6.30935408181637199032909397485, 6.99515174042409879718721289538, 7.88028994378907269348673165738, 8.410578009913751117388162404293, 9.145952634679239803781135370549, 10.03712980714911648556009677479