L(s) = 1 | + (0.866 − 0.5i)3-s + (−2.09 − 1.62i)7-s + (0.499 − 0.866i)9-s + (2.12 + 3.67i)11-s + 5i·13-s + (3.67 − 2.12i)17-s + (1.62 − 2.80i)19-s + (−2.62 − 0.358i)21-s + (−5.19 − 3i)23-s − 0.999i·27-s + 8.48·29-s + (2 + 3.46i)31-s + (3.67 + 2.12i)33-s + (4.33 + 2.5i)37-s + (2.5 + 4.33i)39-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−0.790 − 0.612i)7-s + (0.166 − 0.288i)9-s + (0.639 + 1.10i)11-s + 1.38i·13-s + (0.891 − 0.514i)17-s + (0.371 − 0.644i)19-s + (−0.572 − 0.0782i)21-s + (−1.08 − 0.625i)23-s − 0.192i·27-s + 1.57·29-s + (0.359 + 0.622i)31-s + (0.639 + 0.369i)33-s + (0.711 + 0.410i)37-s + (0.400 + 0.693i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.055277417\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.055277417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.09 + 1.62i)T \) |
good | 11 | \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + (-3.67 + 2.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.62 + 2.80i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 - 4.48iT - 43T^{2} \) |
| 47 | \( 1 + (-1.52 - 0.878i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.67 - 2.12i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.878 - 1.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 6.86i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + (-11.6 + 6.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.378 + 0.655i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (-8.12 + 14.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−9.334310137520425934536144796479, −8.284400871448810088048458666181, −7.47380468117290625508352126024, −6.66757643836762078470481616891, −6.42361512680532801358648312773, −4.82572556693191669754077204888, −4.21037804093505197440224637065, −3.23898944751189214799945577948, −2.22226998903462193636149967781, −1.02885466803301514248937818368,
0.883960705430819541517324968188, 2.45069437209161275042901167248, 3.38399729017221685662413001393, 3.81882071995429178063573532641, 5.33723577460388226509072358295, 5.86585690062007568501359372105, 6.61019288413877330882502371741, 8.037759797085680458850251159726, 8.114705991755156421327610668368, 9.123954067743128633929405260093