L(s) = 1 | + (0.707 − 0.707i)2-s + (1.59 − 0.668i)3-s − 1.00i·4-s + (0.595 − 2.15i)5-s + (0.656 − 1.60i)6-s + (−0.0702 − 0.0702i)7-s + (−0.707 − 0.707i)8-s + (2.10 − 2.13i)9-s + (−1.10 − 1.94i)10-s + 3.25i·11-s + (−0.668 − 1.59i)12-s + (1.84 − 1.84i)13-s − 0.0993·14-s + (−0.489 − 3.84i)15-s − 1.00·16-s + (−3.76 + 3.76i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.922 − 0.386i)3-s − 0.500i·4-s + (0.266 − 0.963i)5-s + (0.268 − 0.654i)6-s + (−0.0265 − 0.0265i)7-s + (−0.250 − 0.250i)8-s + (0.701 − 0.712i)9-s + (−0.348 − 0.615i)10-s + 0.981i·11-s + (−0.193 − 0.461i)12-s + (0.512 − 0.512i)13-s − 0.0265·14-s + (−0.126 − 0.991i)15-s − 0.250·16-s + (−0.913 + 0.913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75021 - 1.94457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75021 - 1.94457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.59 + 0.668i)T \) |
| 5 | \( 1 + (-0.595 + 2.15i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (0.0702 + 0.0702i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.25iT - 11T^{2} \) |
| 13 | \( 1 + (-1.84 + 1.84i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.76 - 3.76i)T - 17iT^{2} \) |
| 23 | \( 1 + (-1.02 - 1.02i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 + (3.00 + 3.00i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 + (2.35 - 2.35i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.871 + 0.871i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.49 - 8.49i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 + (-2.25 - 2.25i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.67iT - 71T^{2} \) |
| 73 | \( 1 + (4.70 - 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.81iT - 79T^{2} \) |
| 83 | \( 1 + (10.5 + 10.5i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.330T + 89T^{2} \) |
| 97 | \( 1 + (3.68 + 3.68i)T + 97iT^{2} \) |
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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
−10.33241239492787261404073978916, −9.654478749299969731845858389081, −8.757788363261234733127804811608, −8.058646404748285682198362712068, −6.87070664330110490415933806563, −5.81581271527291650673811979893, −4.58980003705701652290020380574, −3.79006875614173213337794408165, −2.36828385180400662837740707549, −1.35313503065849172419934258402,
2.33783159813236792457974507235, 3.26994785915438769120785310448, 4.20473621825487821025733182199, 5.44895320405630696835833492919, 6.59399793456274311219032229528, 7.23597566489371637481462611300, 8.382165687834170464767747764895, 9.033774484628149222703523727872, 10.03245803138131524212351214503, 11.00655200620954363190699820838