L(s) = 1 | + (−1.41 + 0.0418i)2-s + (1.99 − 0.118i)4-s + (1.51 + 0.872i)5-s + (1.00 − 2.44i)7-s + (−2.81 + 0.250i)8-s + (−2.17 − 1.17i)10-s + (2.72 − 1.57i)11-s − 3.39i·13-s + (−1.31 + 3.50i)14-s + (3.97 − 0.472i)16-s + (−3.59 + 2.07i)17-s + (2.67 − 4.62i)19-s + (3.12 + 1.56i)20-s + (−3.78 + 2.33i)22-s + (−7.05 − 4.07i)23-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0295i)2-s + (0.998 − 0.0591i)4-s + (0.675 + 0.390i)5-s + (0.380 − 0.924i)7-s + (−0.996 + 0.0886i)8-s + (−0.687 − 0.370i)10-s + (0.822 − 0.474i)11-s − 0.942i·13-s + (−0.352 + 0.935i)14-s + (0.992 − 0.118i)16-s + (−0.871 + 0.503i)17-s + (0.613 − 1.06i)19-s + (0.697 + 0.349i)20-s + (−0.807 + 0.498i)22-s + (−1.47 − 0.849i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.985832 - 0.526612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.985832 - 0.526612i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0418i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.00 + 2.44i)T \) |
good | 5 | \( 1 + (-1.51 - 0.872i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.72 + 1.57i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.39iT - 13T^{2} \) |
| 17 | \( 1 + (3.59 - 2.07i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.67 + 4.62i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.05 + 4.07i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 + (-1.20 - 2.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.00 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.0436iT - 41T^{2} \) |
| 43 | \( 1 - 5.38iT - 43T^{2} \) |
| 47 | \( 1 + (-1.52 + 2.64i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.03 - 3.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.89 + 6.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.01 + 0.588i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.53 + 4.35i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.7iT - 71T^{2} \) |
| 73 | \( 1 + (-14.4 + 8.36i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.56 - 1.48i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + (-4.07 - 2.35i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.9iT - 97T^{2} \) |
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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
−10.32108353992388850727741851912, −9.441177037458616448333060289817, −8.441835618534100026614838741308, −7.83649005516202511787727292165, −6.53346881370066779101784044718, −6.37710563537062971031458858014, −4.83445208756506571000364838079, −3.41729225325196338297644689018, −2.19705488946142593250594118975, −0.819443314662798136103902997637,
1.57795304542673901458819685703, 2.27910573934550401474794494485, 3.95736549464849769815296586465, 5.38499176856565240363292832634, 6.17078481526373877466238822447, 7.08127097483761664166561315487, 8.122322733462041030198039230774, 8.984305612549571285717977832816, 9.466795890489681643636550223840, 10.14769638505343196987083162272