L(s) = 1 | + (0.707 + 0.707i)2-s + (1.08 + 1.34i)3-s + 1.00i·4-s + (0.539 − 2.17i)5-s + (−0.182 + 1.72i)6-s + (2.86 − 2.86i)7-s + (−0.707 + 0.707i)8-s + (−0.629 + 2.93i)9-s + (1.91 − 1.15i)10-s − 4.60i·11-s + (−1.34 + 1.08i)12-s + (1.32 + 1.32i)13-s + 4.05·14-s + (3.51 − 1.63i)15-s − 1.00·16-s + (3.40 + 3.40i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.628 + 0.777i)3-s + 0.500i·4-s + (0.241 − 0.970i)5-s + (−0.0746 + 0.703i)6-s + (1.08 − 1.08i)7-s + (−0.250 + 0.250i)8-s + (−0.209 + 0.977i)9-s + (0.605 − 0.364i)10-s − 1.38i·11-s + (−0.388 + 0.314i)12-s + (0.368 + 0.368i)13-s + 1.08·14-s + (0.906 − 0.422i)15-s − 0.250·16-s + (0.825 + 0.825i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.45608 + 0.852133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45608 + 0.852133i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.08 - 1.34i)T \) |
| 5 | \( 1 + (-0.539 + 2.17i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-2.86 + 2.86i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.60iT - 11T^{2} \) |
| 13 | \( 1 + (-1.32 - 1.32i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.40 - 3.40i)T + 17iT^{2} \) |
| 23 | \( 1 + (2.00 - 2.00i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + (4.28 - 4.28i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.74iT - 41T^{2} \) |
| 43 | \( 1 + (-8.19 - 8.19i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.22 + 5.22i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.96 - 4.96i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.08T + 59T^{2} \) |
| 61 | \( 1 + 0.446T + 61T^{2} \) |
| 67 | \( 1 + (-6.27 + 6.27i)T - 67iT^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 + (6.36 + 6.36i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 + (-3.87 + 3.87i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.07T + 89T^{2} \) |
| 97 | \( 1 + (-12.5 + 12.5i)T - 97iT^{2} \) |
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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
−10.90017914907171012093508485334, −9.878554671864890489822516160851, −8.815431981060034655207484727102, −8.191021650880122731917293066880, −7.58491140664547191902036972861, −5.92721155822431778313777826297, −5.18805152740703376935293443250, −4.15804978392357396965903819503, −3.54824103482353688222196000398, −1.55965727700314204931665143064,
1.83721424565808675016389601289, 2.44483144597876731709490753610, 3.63448883033544686376414197171, 5.11254973235245089321278717895, 5.99469876282860524073324086324, 7.17019722717343647604924514234, 7.79179969699816971127842836937, 9.002368938875564663343108854745, 9.746053748052139596424141543195, 10.83899660970284893160555112717