| L(s) = 1 | − 2.24·2-s + 3-s + 3.04·4-s + 1.69·5-s − 2.24·6-s + 7-s − 2.35·8-s + 9-s − 3.80·10-s − 5.29·11-s + 3.04·12-s − 2.24·14-s + 1.69·15-s − 0.801·16-s − 2.24·17-s − 2.24·18-s − 7.49·19-s + 5.15·20-s + 21-s + 11.8·22-s + 6.76·23-s − 2.35·24-s − 2.13·25-s + 27-s + 3.04·28-s + 7.56·29-s − 3.80·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.52·4-s + 0.756·5-s − 0.917·6-s + 0.377·7-s − 0.833·8-s + 0.333·9-s − 1.20·10-s − 1.59·11-s + 0.880·12-s − 0.600·14-s + 0.436·15-s − 0.200·16-s − 0.544·17-s − 0.529·18-s − 1.71·19-s + 1.15·20-s + 0.218·21-s + 2.53·22-s + 1.41·23-s − 0.481·24-s − 0.427·25-s + 0.192·27-s + 0.576·28-s + 1.40·29-s − 0.694·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 + 9.57T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 6.16T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 0.131T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 0.374T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.390089846746319750200525522129, −7.79593077639991618329221457996, −6.93937348707371645391840774910, −6.33381892144647774200893288278, −5.17471286392607298525133074449, −4.43809654484409714396579303484, −2.84530957505091452253674933720, −2.30748928525852596279724052650, −1.43661095772659531419589913364, 0,
1.43661095772659531419589913364, 2.30748928525852596279724052650, 2.84530957505091452253674933720, 4.43809654484409714396579303484, 5.17471286392607298525133074449, 6.33381892144647774200893288278, 6.93937348707371645391840774910, 7.79593077639991618329221457996, 8.390089846746319750200525522129
