L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (0.299 − 2.21i)5-s − 1.00·6-s + (−3.52 − 3.52i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.35 + 1.77i)10-s + 3.55i·11-s + (0.707 − 0.707i)12-s + (1.85 + 1.85i)13-s + 4.98·14-s + (1.77 − 1.35i)15-s − 1.00·16-s + (−2.03 − 2.03i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.133 − 0.990i)5-s − 0.408·6-s + (−1.33 − 1.33i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.428 + 0.562i)10-s + 1.07i·11-s + (0.204 − 0.204i)12-s + (0.513 + 0.513i)13-s + 1.33·14-s + (0.459 − 0.349i)15-s − 0.250·16-s + (−0.493 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.339647 - 0.480581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339647 - 0.480581i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.299 + 2.21i)T \) |
| 23 | \( 1 + (1.63 + 4.50i)T \) |
good | 7 | \( 1 + (3.52 + 3.52i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.55iT - 11T^{2} \) |
| 13 | \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.03 + 2.03i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 29 | \( 1 + 9.31iT - 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 + (5.97 + 5.97i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.173T + 41T^{2} \) |
| 43 | \( 1 + (-1.69 + 1.69i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.77 + 4.77i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.53 + 4.53i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.64iT - 59T^{2} \) |
| 61 | \( 1 - 5.58iT - 61T^{2} \) |
| 67 | \( 1 + (-6.08 - 6.08i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + (7.81 + 7.81i)T + 73iT^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + (-4.22 + 4.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.69T + 89T^{2} \) |
| 97 | \( 1 + (-9.17 - 9.17i)T + 97iT^{2} \) |
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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
−10.11658474294902142736813276449, −9.231643903896932657460754002009, −8.735938623752306072229288920795, −7.52946311322825643801386221473, −6.85296109321389497430637794856, −5.87460072832896058983253619463, −4.40579704892944419232718239280, −4.01628666283485019955732937482, −2.11917149502579179475692640711, −0.32481311256796470116100126872,
1.97227684592464511325702499691, 3.10534881695219533388818138807, 3.48183249650208235979313579010, 5.74422389535695082761105708452, 6.32066186761856578133361645134, 7.23087508033860500006184448076, 8.523383150560782268339136751804, 8.877986870265432319162231022620, 9.854234312299458377344392080898, 10.71501075117177085855039226644