L(s) = 1 | + (0.541 − 0.393i)2-s + (0.970 + 2.98i)3-s + (−0.479 + 1.47i)4-s + (−1.73 − 1.26i)5-s + (1.70 + 1.23i)6-s + (−0.309 + 0.951i)7-s + (0.735 + 2.26i)8-s + (−5.54 + 4.03i)9-s − 1.43·10-s − 4.87·12-s + (2.04 − 1.48i)13-s + (0.206 + 0.636i)14-s + (2.07 − 6.40i)15-s + (−1.22 − 0.887i)16-s + (−1.44 − 1.05i)17-s + (−1.41 + 4.36i)18-s + ⋯ |
L(s) = 1 | + (0.383 − 0.278i)2-s + (0.560 + 1.72i)3-s + (−0.239 + 0.737i)4-s + (−0.775 − 0.563i)5-s + (0.694 + 0.504i)6-s + (−0.116 + 0.359i)7-s + (0.259 + 0.799i)8-s + (−1.84 + 1.34i)9-s − 0.454·10-s − 1.40·12-s + (0.567 − 0.412i)13-s + (0.0553 + 0.170i)14-s + (0.537 − 1.65i)15-s + (−0.305 − 0.221i)16-s + (−0.351 − 0.255i)17-s + (−0.334 + 1.02i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0631338 + 1.37872i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0631338 + 1.37872i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.541 + 0.393i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.970 - 2.98i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (1.73 + 1.26i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 1.48i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.44 + 1.05i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.07 - 6.39i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + (-0.285 + 0.878i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.42 - 1.76i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.465 - 1.43i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.71 + 5.28i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.42T + 43T^{2} \) |
| 47 | \( 1 + (-1.35 - 4.18i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.539 - 0.392i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.113 - 0.350i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.05 - 2.94i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 0.902T + 67T^{2} \) |
| 71 | \( 1 + (-12.0 - 8.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.48 - 7.64i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.28 + 2.38i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.27 + 2.38i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 + (6.88 - 5.00i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.57671747788848992335690044363, −9.710854184684093454207886486319, −8.838828509625028873338270047392, −8.337672721244907165158052150846, −7.66836192794724520178316142133, −5.75882186354297854092370478135, −4.93584470199631907679834879915, −3.96171427260199578632713914702, −3.69489129518301032851549267108, −2.56209231382159549426057675291,
0.55267068040867357058844035499, 1.85743811863806319970783251504, 3.17730966248824124774393515345, 4.24212687811588125545014116269, 5.67592036968321560429383405417, 6.61941738653337491016443484559, 7.03253344603135703250007801489, 7.79832641445787058860473154715, 8.753701261174721562053365553815, 9.572600443955181561572637080620