L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.5 + 0.866i)7-s − 0.999i·8-s − 3i·11-s + (3 + 1.73i)13-s + (2.59 + 0.500i)14-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)22-s − 5·25-s + (−1.73 − 3i)26-s + (−1.99 − 1.73i)28-s + (7.79 − 4.5i)29-s + (1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.944 + 0.327i)7-s − 0.353i·8-s − 0.904i·11-s + (0.832 + 0.480i)13-s + (0.694 + 0.133i)14-s + (−0.125 + 0.216i)16-s + (−0.319 + 0.553i)22-s − 25-s + (−0.339 − 0.588i)26-s + (−0.377 − 0.327i)28-s + (1.44 − 0.835i)29-s + (0.269 − 0.155i)31-s + (0.153 − 0.0883i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.009882333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009882333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-7.79 + 4.5i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.19 + 9i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.19 + 9i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.822748842148378725047010324629, −8.733632997048166735982068917759, −8.454193631995772923844314163184, −7.24179947223800027832090555189, −6.33619587398958952524540107676, −5.72972261297425682344576767089, −4.16509659605412761393741852829, −3.31040816713884910735546582019, −2.27720520111199977494721844703, −0.68846744758648258084676093827,
1.02943547149077283385947772627, 2.56204477088151213747331425264, 3.73218731241469667515493026067, 4.83892901607449161539398547357, 6.05927797048809356416510584722, 6.57314191007925896585376998668, 7.55046555982897175716525074229, 8.205176436695345061063918330437, 9.294640912068308432228601023894, 9.763148574583642148557030613756