L(s) = 1 | + (1.22 + 0.707i)2-s + (−1.27 − 1.27i)3-s + (0.999 + 1.73i)4-s + (2.08 − 0.796i)5-s + (−0.658 − 2.45i)6-s + (0.691 − 0.691i)7-s + (−8.52e−5 + 2.82i)8-s + 0.236i·9-s + (3.12 + 0.501i)10-s − 2.48i·11-s + (0.931 − 3.47i)12-s + (0.299 − 0.299i)13-s + (1.33 − 0.357i)14-s + (−3.67 − 1.64i)15-s + (−2.00 + 3.46i)16-s + (1.98 + 1.98i)17-s + ⋯ |
L(s) = 1 | + (0.866 + 0.500i)2-s + (−0.734 − 0.734i)3-s + (0.499 + 0.866i)4-s + (0.934 − 0.356i)5-s + (−0.268 − 1.00i)6-s + (0.261 − 0.261i)7-s + (−3.01e−5 + 0.999i)8-s + 0.0789i·9-s + (0.987 + 0.158i)10-s − 0.748i·11-s + (0.268 − 1.00i)12-s + (0.0830 − 0.0830i)13-s + (0.356 − 0.0956i)14-s + (−0.947 − 0.424i)15-s + (−0.500 + 0.866i)16-s + (0.482 + 0.482i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07378 - 0.134280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07378 - 0.134280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 5 | \( 1 + (-2.08 + 0.796i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + (1.27 + 1.27i)T + 3iT^{2} \) |
| 7 | \( 1 + (-0.691 + 0.691i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.48iT - 11T^{2} \) |
| 13 | \( 1 + (-0.299 + 0.299i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.98 - 1.98i)T + 17iT^{2} \) |
| 23 | \( 1 + (-3.56 - 3.56i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.50iT - 29T^{2} \) |
| 31 | \( 1 + 1.55iT - 31T^{2} \) |
| 37 | \( 1 + (-1.57 - 1.57i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + (-0.612 - 0.612i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.25 - 7.25i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.698 + 0.698i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + (9.08 - 9.08i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.70iT - 71T^{2} \) |
| 73 | \( 1 + (1.79 - 1.79i)T - 73iT^{2} \) |
| 79 | \( 1 - 0.823T + 79T^{2} \) |
| 83 | \( 1 + (-0.768 - 0.768i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + (5.43 + 5.43i)T + 97iT^{2} \) |
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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
−11.59820463466114368265583544394, −10.77374922358498506346749629389, −9.438599589526190902725089477003, −8.264587517069716361633385770160, −7.30299053534543396414334593430, −6.19526413716310244699661135504, −5.80713331842053639282698197693, −4.72867541001664992429176524721, −3.20030155742491726192753925124, −1.45983130749709916125100908187,
1.84822515176138973980845249938, 3.19902569442130128593347679498, 4.77654901017572934141736641118, 5.19798373806770462847902684876, 6.24001062725099981219776515554, 7.21347999648618167624995391938, 9.042986288330194201039953739396, 10.03826589746805402186634178872, 10.47819328407576586728459262322, 11.34371929832973671560582353893