| L(s) = 1 | − 0.414i·2-s − i·3-s + 1.82·4-s − 0.414·6-s + 1.41i·7-s − 1.58i·8-s − 9-s + 6.24·11-s − 1.82i·12-s + 0.585i·13-s + 0.585·14-s + 3·16-s + 6.82i·17-s + 0.414i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.292i·2-s − 0.577i·3-s + 0.914·4-s − 0.169·6-s + 0.534i·7-s − 0.560i·8-s − 0.333·9-s + 1.88·11-s − 0.527i·12-s + 0.162i·13-s + 0.156·14-s + 0.750·16-s + 1.65i·17-s + 0.0976i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.374757820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374757820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 - 0.585iT - 13T^{2} \) |
| 17 | \( 1 - 6.82iT - 17T^{2} \) |
| 23 | \( 1 - 3.65iT - 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 0.585iT - 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 3.75iT - 43T^{2} \) |
| 47 | \( 1 - 3.65iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.65iT - 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 7.17iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 18.2iT - 97T^{2} \) |
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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
−9.352147342749707751922802595487, −8.801028995391613260057329493952, −7.72857533819501736901046728781, −7.00111876429057317118707877490, −6.19758061829183368564606451740, −5.73984860669607466817563695203, −4.07772798978149108396007997038, −3.33634389864119075962357501136, −1.99480225568683926891243326483, −1.37338966499829212168072227509,
1.12656860184657749837962393055, 2.56906719787769126352413327608, 3.60234054773729432302132623839, 4.48779188255073937687651814638, 5.54639009073670475167049081580, 6.45262407957720257647052304546, 7.09088503208853477245548305300, 7.80103992976662685706692277025, 9.057021153050678480854372836365, 9.401472788146453619854009513853
