L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (−1.29 + 1.81i)5-s − 1.00·6-s + (−0.728 − 0.728i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.367 + 2.20i)10-s − 4.80·11-s + (−0.707 + 0.707i)12-s + (0.531 + 0.531i)13-s − 1.03·14-s + (2.20 − 0.367i)15-s − 1.00·16-s + (−3.72 − 3.72i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.581 + 0.813i)5-s − 0.408·6-s + (−0.275 − 0.275i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.116 + 0.697i)10-s − 1.44·11-s + (−0.204 + 0.204i)12-s + (0.147 + 0.147i)13-s − 0.275·14-s + (0.569 − 0.0948i)15-s − 0.250·16-s + (−0.904 − 0.904i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00218473 + 0.00519485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00218473 + 0.00519485i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.29 - 1.81i)T \) |
| 19 | \( 1 + (2.90 - 3.24i)T \) |
good | 7 | \( 1 + (0.728 + 0.728i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + (-0.531 - 0.531i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.72 + 3.72i)T + 17iT^{2} \) |
| 23 | \( 1 + (4.07 - 4.07i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.494T + 29T^{2} \) |
| 31 | \( 1 - 8.62iT - 31T^{2} \) |
| 37 | \( 1 + (-5.47 + 5.47i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.82iT - 41T^{2} \) |
| 43 | \( 1 + (3.04 - 3.04i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.910 - 0.910i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.53 - 3.53i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 + (-9.19 + 9.19i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.06iT - 71T^{2} \) |
| 73 | \( 1 + (-3.31 + 3.31i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.77T + 79T^{2} \) |
| 83 | \( 1 + (3.57 - 3.57i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + (-2.81 + 2.81i)T - 97iT^{2} \) |
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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
−10.69926625732343971349397124406, −9.620835660875140613170696567044, −8.211374945153900454767527740096, −7.35935822109705020519112878038, −6.52747319634078099693437429276, −5.51984112973059459114140750379, −4.40377541596798577181444931057, −3.25230625540114771749793172036, −2.17131990697141187011829623520, −0.00263088369157415195831078720,
2.57846328624266421113887293600, 4.08862849583636702241095987383, 4.72276548341221147622890266208, 5.72079071577174092515363633519, 6.54402450127290885317087292650, 7.915034131188319526453561611679, 8.392422641969562982880866205074, 9.449714831877030528329640116760, 10.55360976554263578791046331210, 11.32234450347974121282304871838