L(s) = 1 | + (−0.159 + 0.125i)2-s + (0.814 − 0.580i)3-s + (−0.461 + 1.90i)4-s + (−3.31 − 0.639i)5-s + (−0.0570 + 0.194i)6-s + (1.87 − 1.86i)7-s + (−0.333 − 0.729i)8-s + (0.327 − 0.945i)9-s + (0.608 − 0.313i)10-s + (3.22 − 4.09i)11-s + (0.728 + 1.81i)12-s + (−1.15 + 1.79i)13-s + (−0.0662 + 0.531i)14-s + (−3.07 + 1.40i)15-s + (−3.33 − 1.72i)16-s + (3.59 − 3.42i)17-s + ⋯ |
L(s) = 1 | + (−0.112 + 0.0885i)2-s + (0.470 − 0.334i)3-s + (−0.230 + 0.951i)4-s + (−1.48 − 0.286i)5-s + (−0.0232 + 0.0793i)6-s + (0.710 − 0.703i)7-s + (−0.117 − 0.257i)8-s + (0.109 − 0.315i)9-s + (0.192 − 0.0992i)10-s + (0.972 − 1.23i)11-s + (0.210 + 0.524i)12-s + (−0.319 + 0.497i)13-s + (−0.0176 + 0.142i)14-s + (−0.794 + 0.362i)15-s + (−0.834 − 0.430i)16-s + (0.871 − 0.830i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08871 - 0.513145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08871 - 0.513145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (-1.87 + 1.86i)T \) |
| 23 | \( 1 + (-0.195 + 4.79i)T \) |
good | 2 | \( 1 + (0.159 - 0.125i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (3.31 + 0.639i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-3.22 + 4.09i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.15 - 1.79i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.59 + 3.42i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.34 - 2.23i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-5.28 - 1.55i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.552 + 5.78i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-9.27 - 3.21i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (3.67 + 3.18i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (10.7 + 4.92i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (9.15 - 5.28i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.978 + 0.0466i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.02 + 3.93i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (4.91 - 6.89i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (0.244 - 0.611i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.59 - 11.0i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-7.21 - 1.75i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-10.6 - 0.508i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (7.36 + 8.49i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (8.61 - 0.822i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (2.77 - 3.20i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.33125820665769840714376411804, −9.762348321284801497419819746465, −8.607158458291508568126457090607, −8.164173239077551346802592122978, −7.51084623159454933755243772570, −6.64508137289342960285346939585, −4.72414003031492722014296204760, −3.90400399297596372564986280628, −3.12996286559581875751188878570, −0.823107412714376704578833985918,
1.56751374316146753325409698197, 3.22360205359666127634965752350, 4.44312289017259184497169378983, 5.13692865593149313319030871711, 6.58337843542008696830466269932, 7.71821073803068899481271978942, 8.363294967997029669729665566503, 9.439323117524387237295652687441, 10.08564923013984095088537458143, 11.19287064520143475077355564653