L(s) = 1 | + (−1.41 + 0.0989i)2-s + (1.98 − 0.279i)4-s + (−0.795 − 0.459i)5-s + (0.866 − 0.5i)7-s + (−2.76 + 0.589i)8-s + (1.16 + 0.569i)10-s + (−0.582 − 1.00i)11-s + (0.0273 − 0.0472i)13-s + (−1.17 + 0.791i)14-s + (3.84 − 1.10i)16-s − 3.29i·17-s − 0.455i·19-s + (−1.70 − 0.687i)20-s + (0.921 + 1.36i)22-s + (−1.77 + 3.08i)23-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0699i)2-s + (0.990 − 0.139i)4-s + (−0.355 − 0.205i)5-s + (0.327 − 0.188i)7-s + (−0.978 + 0.208i)8-s + (0.369 + 0.180i)10-s + (−0.175 − 0.304i)11-s + (0.00757 − 0.0131i)13-s + (−0.313 + 0.211i)14-s + (0.961 − 0.276i)16-s − 0.799i·17-s − 0.104i·19-s + (−0.381 − 0.153i)20-s + (0.196 + 0.291i)22-s + (−0.370 + 0.642i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0916 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0916 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.465996 - 0.510854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.465996 - 0.510854i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0989i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.795 + 0.459i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.582 + 1.00i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0273 + 0.0472i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.29iT - 17T^{2} \) |
| 19 | \( 1 + 0.455iT - 19T^{2} \) |
| 23 | \( 1 + (1.77 - 3.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.58 + 4.37i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.03 + 4.06i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 + (-5.27 - 3.04i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.64 - 3.25i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.82 + 4.88i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + (0.744 - 1.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.35 + 11.0i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.83 + 3.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + (-10.9 + 6.31i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.56 + 11.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.95iT - 89T^{2} \) |
| 97 | \( 1 + (4.26 + 7.38i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.945719610070966092224608636800, −9.339344263706652520241006858620, −8.222730333822404737343527114052, −7.84761827963126513825373594883, −6.81848855701266354912971097214, −5.87861541259994670088627294460, −4.72121714445016029638683259411, −3.35515501422870411949087268735, −2.05998775858926911395528543325, −0.49478312578478403560441120316,
1.48682097354735599427043265357, 2.74961242634849041281715890955, 3.96097734432163583044474672689, 5.37024525877203057711019635075, 6.44112074141400544490247085999, 7.29846789717281763638080414488, 8.097859716293894068758529821503, 8.802511097471285986987065032291, 9.685527505086371091347055505884, 10.68472465857564727480289161471