| L(s) = 1 | − 1.24·2-s − 0.445·4-s + 2.69·5-s − 0.198·7-s + 3.04·8-s − 3.35·10-s − 0.137·11-s − 2.35·13-s + 0.246·14-s − 2.91·16-s − 5.96·17-s − 7.35·19-s − 1.19·20-s + 0.170·22-s − 1.19·23-s + 2.24·25-s + 2.93·26-s + 0.0881·28-s + 4.96·29-s − 6.33·31-s − 2.46·32-s + 7.43·34-s − 0.533·35-s + 10.4·37-s + 9.17·38-s + 8.20·40-s + 5.78·41-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.222·4-s + 1.20·5-s − 0.0748·7-s + 1.07·8-s − 1.06·10-s − 0.0413·11-s − 0.653·13-s + 0.0660·14-s − 0.727·16-s − 1.44·17-s − 1.68·19-s − 0.267·20-s + 0.0364·22-s − 0.249·23-s + 0.449·25-s + 0.576·26-s + 0.0166·28-s + 0.921·29-s − 1.13·31-s − 0.436·32-s + 1.27·34-s − 0.0901·35-s + 1.72·37-s + 1.48·38-s + 1.29·40-s + 0.904·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 0.198T + 7T^{2} \) |
| 11 | \( 1 + 0.137T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 7.35T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 + 0.554T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.344735894165901055470776844051, −8.588927852169115695084820065735, −7.82026787392205929246678407221, −6.71975756277465805020737376580, −6.10175107193518534328878870620, −4.90456617860547474139793210622, −4.23521119048091748357930058478, −2.50076913255226420823801534364, −1.70791258832463289433855915367, 0,
1.70791258832463289433855915367, 2.50076913255226420823801534364, 4.23521119048091748357930058478, 4.90456617860547474139793210622, 6.10175107193518534328878870620, 6.71975756277465805020737376580, 7.82026787392205929246678407221, 8.588927852169115695084820065735, 9.344735894165901055470776844051
