| L(s) = 1 | + 0.782·2-s + 1.34·3-s − 1.38·4-s − 0.742·5-s + 1.05·6-s − 2.34·7-s − 2.65·8-s − 1.18·9-s − 0.581·10-s − 11-s − 1.86·12-s − 2.09·13-s − 1.83·14-s − 15-s + 0.699·16-s − 4.14·17-s − 0.928·18-s + 3.60·19-s + 1.02·20-s − 3.16·21-s − 0.782·22-s − 0.776·23-s − 3.57·24-s − 4.44·25-s − 1.63·26-s − 5.63·27-s + 3.25·28-s + ⋯ |
| L(s) = 1 | + 0.553·2-s + 0.777·3-s − 0.693·4-s − 0.332·5-s + 0.430·6-s − 0.887·7-s − 0.937·8-s − 0.395·9-s − 0.183·10-s − 0.301·11-s − 0.539·12-s − 0.580·13-s − 0.490·14-s − 0.258·15-s + 0.174·16-s − 1.00·17-s − 0.218·18-s + 0.826·19-s + 0.230·20-s − 0.689·21-s − 0.166·22-s − 0.161·23-s − 0.729·24-s − 0.889·25-s − 0.321·26-s − 1.08·27-s + 0.615·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.782T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 0.742T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + 0.776T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 - 8.69T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 8.90T + 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 + 0.786T + 83T^{2} \) |
| 89 | \( 1 + 4.13T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−9.606242139747453298522566391244, −9.415688117774351761006236537744, −8.361757666374228673264251958380, −7.60908309911086684142827933705, −6.36743487272197385130949725936, −5.42708133505674306197001399466, −4.31721828151194991557550212682, −3.40615673315827058921822313712, −2.56562691549407031890805347774, 0,
2.56562691549407031890805347774, 3.40615673315827058921822313712, 4.31721828151194991557550212682, 5.42708133505674306197001399466, 6.36743487272197385130949725936, 7.60908309911086684142827933705, 8.361757666374228673264251958380, 9.415688117774351761006236537744, 9.606242139747453298522566391244
