| L(s) = 1 | − 1.05·2-s − 3-s − 0.890·4-s + 2.17·5-s + 1.05·6-s + 3.04·8-s + 9-s − 2.29·10-s − 2.62·11-s + 0.890·12-s − 3.30·13-s − 2.17·15-s − 1.42·16-s − 2.96·17-s − 1.05·18-s + 0.697·19-s − 1.93·20-s + 2.76·22-s + 23-s − 3.04·24-s − 0.264·25-s + 3.48·26-s − 27-s + 7.31·29-s + 2.29·30-s + 5.60·31-s − 4.58·32-s + ⋯ |
| L(s) = 1 | − 0.744·2-s − 0.577·3-s − 0.445·4-s + 0.973·5-s + 0.430·6-s + 1.07·8-s + 0.333·9-s − 0.724·10-s − 0.790·11-s + 0.256·12-s − 0.917·13-s − 0.561·15-s − 0.356·16-s − 0.720·17-s − 0.248·18-s + 0.160·19-s − 0.433·20-s + 0.589·22-s + 0.208·23-s − 0.621·24-s − 0.0529·25-s + 0.683·26-s − 0.192·27-s + 1.35·29-s + 0.418·30-s + 1.00·31-s − 0.810·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7750859986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7750859986\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 0.697T + 19T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 - 2.28T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.48T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.705T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 0.333T + 89T^{2} \) |
| 97 | \( 1 - 0.124T + 97T^{2} \) |
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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
−8.569118752759111215275319635165, −8.097781076750739873256111172444, −7.07210998269816753830922291291, −6.53043785892325795312855853424, −5.38346579134180212653267153611, −5.03992850171119329975317930736, −4.17178921218129550888082215515, −2.72832929673530371904236302188, −1.82610538665401551553390315131, −0.60451917206128996640654774511,
0.60451917206128996640654774511, 1.82610538665401551553390315131, 2.72832929673530371904236302188, 4.17178921218129550888082215515, 5.03992850171119329975317930736, 5.38346579134180212653267153611, 6.53043785892325795312855853424, 7.07210998269816753830922291291, 8.097781076750739873256111172444, 8.569118752759111215275319635165
