| L(s) = 1 | − 1.77·2-s + 3-s + 1.16·4-s + 0.681·5-s − 1.77·6-s + 7-s + 1.48·8-s + 9-s − 1.21·10-s + 0.590·11-s + 1.16·12-s − 1.77·14-s + 0.681·15-s − 4.97·16-s − 6.29·17-s − 1.77·18-s − 3.58·19-s + 0.794·20-s + 21-s − 1.05·22-s + 3.68·23-s + 1.48·24-s − 4.53·25-s + 27-s + 1.16·28-s − 6.45·29-s − 1.21·30-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.577·3-s + 0.582·4-s + 0.304·5-s − 0.726·6-s + 0.377·7-s + 0.524·8-s + 0.333·9-s − 0.383·10-s + 0.177·11-s + 0.336·12-s − 0.475·14-s + 0.176·15-s − 1.24·16-s − 1.52·17-s − 0.419·18-s − 0.822·19-s + 0.177·20-s + 0.218·21-s − 0.223·22-s + 0.768·23-s + 0.302·24-s − 0.907·25-s + 0.192·27-s + 0.220·28-s − 1.19·29-s − 0.221·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175646898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175646898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 - 0.681T + 5T^{2} \) |
| 11 | \( 1 - 0.590T + 11T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 0.157T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.792065207168911238961261290089, −7.982210196993240273790994986110, −7.36434197688132555436662235803, −6.64885492333939052648594185614, −5.72067889017350726635244667529, −4.50832186886212748710950308306, −4.04540473378088844063466412484, −2.49257214919019554627654455739, −1.96844671485806211921379318932, −0.75410487873716902136650210487,
0.75410487873716902136650210487, 1.96844671485806211921379318932, 2.49257214919019554627654455739, 4.04540473378088844063466412484, 4.50832186886212748710950308306, 5.72067889017350726635244667529, 6.64885492333939052648594185614, 7.36434197688132555436662235803, 7.982210196993240273790994986110, 8.792065207168911238961261290089
