L(s) = 1 | + (1.36 − 1.36i)3-s + (−0.866 − 0.5i)5-s − 2.73i·9-s + i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (1.36 − 1.36i)37-s + (−1.36 + 2.36i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯ |
L(s) = 1 | + (1.36 − 1.36i)3-s + (−0.866 − 0.5i)5-s − 2.73i·9-s + i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (1.36 − 1.36i)37-s + (−1.36 + 2.36i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.307318746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.307318746\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
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
−9.738784294800882116945047082011, −9.065960405605261693077830906208, −8.318036531506375984359046614845, −7.54060717930936533961767526011, −7.19248134360475378628894512528, −6.05658600125537919109297168332, −4.51015432566838211828285673862, −3.56159649848261142436627024358, −2.47176325094174984925014367806, −1.30898036881670797950025826654,
2.50056518450525596810840107745, 3.32592014572024620001501582188, 4.03988941904665264019732286370, 4.88540701164703095623888673719, 6.22094143731718373599574831723, 7.63527527492659467278258616601, 8.128681149053273622312744640116, 8.860926954601703184616761703387, 9.668442940790191879555065636418, 10.50776876473902676792622230991