| L(s) = 1 | − 2-s − 2.34·3-s + 4-s − 2.76·5-s + 2.34·6-s + 1.79·7-s − 8-s + 2.49·9-s + 2.76·10-s − 11-s − 2.34·12-s − 5.14·13-s − 1.79·14-s + 6.49·15-s + 16-s − 5.04·17-s − 2.49·18-s − 2.16·19-s − 2.76·20-s − 4.21·21-s + 22-s − 3.74·23-s + 2.34·24-s + 2.66·25-s + 5.14·26-s + 1.18·27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.35·3-s + 0.5·4-s − 1.23·5-s + 0.957·6-s + 0.679·7-s − 0.353·8-s + 0.831·9-s + 0.875·10-s − 0.301·11-s − 0.676·12-s − 1.42·13-s − 0.480·14-s + 1.67·15-s + 0.250·16-s − 1.22·17-s − 0.588·18-s − 0.496·19-s − 0.619·20-s − 0.919·21-s + 0.213·22-s − 0.780·23-s + 0.478·24-s + 0.533·25-s + 1.00·26-s + 0.227·27-s + 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002674614349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002674614349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 7.63T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 + 7.75T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 0.387T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.233321156374701918117079168940, −7.59530294561373909519413192445, −7.08609998148895595350261600793, −6.28878553540026158567953432535, −5.42296757597640231523283603974, −4.65584714447740189740474862913, −4.12146743940567621742025137318, −2.73403419692117567270769930511, −1.66243168322504853758371430174, −0.03380945317968314573229901984,
0.03380945317968314573229901984, 1.66243168322504853758371430174, 2.73403419692117567270769930511, 4.12146743940567621742025137318, 4.65584714447740189740474862913, 5.42296757597640231523283603974, 6.28878553540026158567953432535, 7.08609998148895595350261600793, 7.59530294561373909519413192445, 8.233321156374701918117079168940
