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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0788471 - 0.229152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0788471 - 0.229152i\) |
\(L(\frac{3}{2})\) |
|
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
−11.09102055545469856709326755637, −10.04678555425335439041388444647, −9.804486599813756921522399171141, −8.400016251607443526263220056830, −7.73726650929331890216209127313, −7.09470675179528863100960790530, −5.89618932258077967647120110086, −4.78849517007756899274998765658, −3.52453243666616173872466582710, −2.43495360141654852813626981822,
0.14336639912651409025233632331, 1.93730779021921061175212371712, 3.09678994177664536176657466006, 4.35110296535853345381372354861, 5.46062372940993664647818250631, 6.90020834977339454109445618484, 7.78508007536442461012002463332, 8.578685695952325229402552263336, 9.071180250463829314777410766388, 10.26578343823567061560179337965