| L(s) = 1 | − 1.58·2-s − 1.33·3-s + 0.527·4-s − 0.921·5-s + 2.12·6-s + 2.34·8-s − 1.21·9-s + 1.46·10-s − 4.89·11-s − 0.705·12-s − 2.00·13-s + 1.23·15-s − 4.77·16-s + 4.36·17-s + 1.92·18-s + 19-s − 0.485·20-s + 7.78·22-s − 2.75·23-s − 3.13·24-s − 4.15·25-s + 3.19·26-s + 5.63·27-s − 1.04·29-s − 1.95·30-s − 7.63·31-s + 2.91·32-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.772·3-s + 0.263·4-s − 0.411·5-s + 0.868·6-s + 0.827·8-s − 0.403·9-s + 0.463·10-s − 1.47·11-s − 0.203·12-s − 0.557·13-s + 0.318·15-s − 1.19·16-s + 1.05·17-s + 0.453·18-s + 0.229·19-s − 0.108·20-s + 1.65·22-s − 0.574·23-s − 0.639·24-s − 0.830·25-s + 0.626·26-s + 1.08·27-s − 0.194·29-s − 0.357·30-s − 1.37·31-s + 0.514·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2791330666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2791330666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 + 0.921T + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 + 1.04T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 - 9.80T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 5.56T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 - 3.89T + 97T^{2} \) |
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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
−10.19281059337693506412019302711, −9.304589094246752447137580373742, −8.254793107958062692457367988978, −7.77910828715247042589290902655, −6.97475092858580311520200175006, −5.54339477559798626531402929130, −5.13102505674063361544111235641, −3.70846904263995147542102589037, −2.22616695377258739916365377229, −0.49143273193612497414370253394,
0.49143273193612497414370253394, 2.22616695377258739916365377229, 3.70846904263995147542102589037, 5.13102505674063361544111235641, 5.54339477559798626531402929130, 6.97475092858580311520200175006, 7.77910828715247042589290902655, 8.254793107958062692457367988978, 9.304589094246752447137580373742, 10.19281059337693506412019302711
