L(s) = 1 | + 2-s + 0.524·3-s + 4-s + 0.579·5-s + 0.524·6-s + 1.74·7-s + 8-s − 2.72·9-s + 0.579·10-s − 4.24·11-s + 0.524·12-s − 1.10·13-s + 1.74·14-s + 0.304·15-s + 16-s + 0.907·17-s − 2.72·18-s − 4.36·19-s + 0.579·20-s + 0.917·21-s − 4.24·22-s − 5.04·23-s + 0.524·24-s − 4.66·25-s − 1.10·26-s − 3.00·27-s + 1.74·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.302·3-s + 0.5·4-s + 0.259·5-s + 0.214·6-s + 0.660·7-s + 0.353·8-s − 0.908·9-s + 0.183·10-s − 1.27·11-s + 0.151·12-s − 0.306·13-s + 0.467·14-s + 0.0785·15-s + 0.250·16-s + 0.220·17-s − 0.642·18-s − 1.00·19-s + 0.129·20-s + 0.200·21-s − 0.904·22-s − 1.05·23-s + 0.107·24-s − 0.932·25-s − 0.216·26-s − 0.578·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 0.524T + 3T^{2} \) |
| 5 | \( 1 - 0.579T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 0.907T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 - 2.42T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 - 5.50T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 6.83T + 97T^{2} \) |
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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
−8.131035726601558190929321838779, −7.44925574649819497073464902874, −6.43319574561053630238166507883, −5.62874047558129172401003688548, −5.22551857112836646915027155775, −4.27852378248653973795759650429, −3.42248129819471588401240521315, −2.41713229785930014000230726785, −1.93110098744198250204289805085, 0,
1.93110098744198250204289805085, 2.41713229785930014000230726785, 3.42248129819471588401240521315, 4.27852378248653973795759650429, 5.22551857112836646915027155775, 5.62874047558129172401003688548, 6.43319574561053630238166507883, 7.44925574649819497073464902874, 8.131035726601558190929321838779