L(s) = 1 | + (0.809 − 1.15i)2-s + (−0.690 − 1.87i)4-s + (1.03 − 1.78i)5-s + (2.39 − 1.13i)7-s + (−2.73 − 0.718i)8-s + (−1.23 − 2.64i)10-s + (0.982 + 1.70i)11-s + 4.20·13-s + (0.623 − 3.68i)14-s + (−3.04 + 2.59i)16-s + (−3.09 + 1.78i)17-s + (−4.36 − 2.52i)19-s + (−4.06 − 0.703i)20-s + (2.76 + 0.237i)22-s + (−5.31 − 3.06i)23-s + ⋯ |
L(s) = 1 | + (0.572 − 0.820i)2-s + (−0.345 − 0.938i)4-s + (0.461 − 0.799i)5-s + (0.904 − 0.427i)7-s + (−0.967 − 0.253i)8-s + (−0.391 − 0.835i)10-s + (0.296 + 0.512i)11-s + 1.16·13-s + (0.166 − 0.986i)14-s + (−0.761 + 0.647i)16-s + (−0.751 + 0.434i)17-s + (−1.00 − 0.578i)19-s + (−0.909 − 0.157i)20-s + (0.590 + 0.0506i)22-s + (−1.10 − 0.639i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09965 - 1.78286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09965 - 1.78286i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 1.15i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.39 + 1.13i)T \) |
good | 5 | \( 1 + (-1.03 + 1.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.982 - 1.70i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 17 | \( 1 + (3.09 - 1.78i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.36 + 2.52i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.31 + 3.06i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.34iT - 29T^{2} \) |
| 31 | \( 1 + (-3.42 - 5.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.56 - 2.05i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.45iT - 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 + (4.88 - 8.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.2 + 6.48i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.17 - 3.56i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.03 - 1.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.77 - 4.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.44iT - 71T^{2} \) |
| 73 | \( 1 + (10.7 - 6.18i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.778 - 0.449i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.58iT - 83T^{2} \) |
| 89 | \( 1 + (-12.6 - 7.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.550iT - 97T^{2} \) |
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
−10.77562585530643131761777093316, −9.968336769363718902426099051585, −8.870523072438165294675092897674, −8.316691408261240604924873691641, −6.65688820091771819931632239329, −5.75773270089211316659301548526, −4.55561351304402499452898976871, −4.11461228861221669775204339549, −2.25940636024793470410683297295, −1.19673698376693689806035148649,
2.19259285391887318752257735878, 3.56497306425416190231003552693, 4.62620548270520973196168004707, 5.96919132518305047927707636649, 6.26788582759397386313837487030, 7.51961568310275737088997073995, 8.420891120355433865694602503129, 9.084476036758748366280108759313, 10.48288525862848699075062433874, 11.30948824955037197321562466445