L(s) = 1 | + (−1.73 − 5.38i)2-s + (−25.9 + 18.6i)4-s − 91.5i·5-s + 158. i·7-s + (145. + 107. i)8-s + (−493 + 158. i)10-s − 580.·11-s − 166·13-s + (853. − 274. i)14-s + (327. − 970. i)16-s + 829. i·17-s + 671. i·19-s + (1.70e3 + 2.38e3i)20-s + (1.00e3 + 3.12e3i)22-s + 3.85e3·23-s + ⋯ |
L(s) = 1 | + (−0.306 − 0.951i)2-s + (−0.812 + 0.582i)4-s − 1.63i·5-s + 1.22i·7-s + (0.803 + 0.594i)8-s + (−1.55 + 0.501i)10-s − 1.44·11-s − 0.272·13-s + (1.16 − 0.374i)14-s + (0.320 − 0.947i)16-s + 0.695i·17-s + 0.426i·19-s + (0.954 + 1.33i)20-s + (0.442 + 1.37i)22-s + 1.52·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.651789 + 0.209637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.651789 + 0.209637i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 + 5.38i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 91.5iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 158. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 580.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 166T + 3.71e5T^{2} \) |
| 17 | \( 1 - 829. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 671. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.41e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.33e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.53e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.01e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 3.00e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.72e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.10e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.01e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.86e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.37e4T + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.75803393606929301768992890281, −12.04109533841478495276197146386, −10.78489470466492591200738167721, −9.512273713445341251802581407611, −8.702323969995413705393106513413, −7.967687086925914964170597447543, −5.46099797467672802658819417288, −4.72661546642776125940035274986, −2.80451723785192321413831739066, −1.33690246075556839003864221758,
0.30877576698172027352091276740, 2.88071851800310368833992851613, 4.57721111555965388378629222644, 6.13141339692114794077444533977, 7.35779739041483077347500254128, 7.61069747322674671738057090547, 9.532754245393054792063335209639, 10.50771011576599196460934211529, 11.07102066859651369138776038172, 13.23950826850618893533996589546