| L(s) = 1 | − 1.61·2-s + 1.61·3-s + 0.618·4-s − 2.85·5-s − 2.61·6-s + 2.23·7-s + 2.23·8-s − 0.381·9-s + 4.61·10-s − 3.61·11-s + 1.00·12-s + 4.23·13-s − 3.61·14-s − 4.61·15-s − 4.85·16-s − 6.61·17-s + 0.618·18-s + 1.85·19-s − 1.76·20-s + 3.61·21-s + 5.85·22-s + 3.23·23-s + 3.61·24-s + 3.14·25-s − 6.85·26-s − 5.47·27-s + 1.38·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.934·3-s + 0.309·4-s − 1.27·5-s − 1.06·6-s + 0.845·7-s + 0.790·8-s − 0.127·9-s + 1.46·10-s − 1.09·11-s + 0.288·12-s + 1.17·13-s − 0.966·14-s − 1.19·15-s − 1.21·16-s − 1.60·17-s + 0.145·18-s + 0.425·19-s − 0.394·20-s + 0.789·21-s + 1.24·22-s + 0.674·23-s + 0.738·24-s + 0.629·25-s − 1.34·26-s − 1.05·27-s + 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 0.291T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + 8.70T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−9.453766446592641792838607570399, −8.583496549012541642589046674846, −8.318697189967849552966517337284, −7.74168767004205784245139163213, −6.83918953864746637229187488986, −5.11913862042912634519027874535, −4.19535437260133487445131990904, −3.10535176000416497470360599770, −1.73987673421271270695939834898, 0,
1.73987673421271270695939834898, 3.10535176000416497470360599770, 4.19535437260133487445131990904, 5.11913862042912634519027874535, 6.83918953864746637229187488986, 7.74168767004205784245139163213, 8.318697189967849552966517337284, 8.583496549012541642589046674846, 9.453766446592641792838607570399
