L(s) = 1 | − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯ |
L(s) = 1 | − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7886763449\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7886763449\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.04854837437942496760166223865, −9.647044327939144600882331096177, −8.368606414729696413023233107418, −7.29596142280420772880598090374, −6.59390574852220492798975328233, −5.76192932390643374241656158249, −4.97351158894988733359554889810, −3.97475509500580425100787307385, −2.49730313996834084159477406008, −0.888355604703299204175892473574,
1.76567963821866631400230613918, 2.80834934473539215359122660250, 4.42751356559661842221418965394, 5.55201673953473326287727927844, 5.96257813594915694726073057534, 6.46429044551159361631351075448, 8.043006352600327529359166118573, 8.731718541648375726483639070846, 9.698767793115177574493064887325, 10.29899130671281585511177224199