L(s) = 1 | + (0.945 − 1.45i)3-s + (−1.69 + 1.69i)5-s − 7-s + (−1.21 − 2.74i)9-s + (0.445 + 0.445i)11-s + (−1.51 + 1.51i)13-s + (0.856 + 4.06i)15-s + 1.39i·17-s + (−0.965 − 0.965i)19-s + (−0.945 + 1.45i)21-s + 6.04i·23-s − 0.748i·25-s + (−5.12 − 0.839i)27-s + (4.93 + 4.93i)29-s + 0.470i·31-s + ⋯ |
L(s) = 1 | + (0.546 − 0.837i)3-s + (−0.758 + 0.758i)5-s − 0.377·7-s + (−0.403 − 0.914i)9-s + (0.134 + 0.134i)11-s + (−0.419 + 0.419i)13-s + (0.221 + 1.04i)15-s + 0.337i·17-s + (−0.221 − 0.221i)19-s + (−0.206 + 0.316i)21-s + 1.26i·23-s − 0.149i·25-s + (−0.986 − 0.161i)27-s + (0.915 + 0.915i)29-s + 0.0844i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.008033798\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.008033798\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.945 + 1.45i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (1.69 - 1.69i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.445 - 0.445i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.51 - 1.51i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.39iT - 17T^{2} \) |
| 19 | \( 1 + (0.965 + 0.965i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.04iT - 23T^{2} \) |
| 29 | \( 1 + (-4.93 - 4.93i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.470iT - 31T^{2} \) |
| 37 | \( 1 + (-1.47 - 1.47i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.35T + 41T^{2} \) |
| 43 | \( 1 + (8.97 - 8.97i)T - 43iT^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + (3.42 - 3.42i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.61 - 6.61i)T + 59iT^{2} \) |
| 61 | \( 1 + (8.04 - 8.04i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.38 + 4.38i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 + 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 16.5iT - 79T^{2} \) |
| 83 | \( 1 + (9.24 - 9.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.25T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−9.645844334725851085512598330423, −8.933101701696920485881080630541, −7.993278802215272470222933236465, −7.34304857014438931004446969383, −6.75297142256589281851427136624, −5.95017402070865973909496168335, −4.53462387247054356878545733441, −3.43972633294426141876831381408, −2.81179868778434208109175545424, −1.46734475777539083029434181304,
0.38854416491551520225231313055, 2.37074384278330430292016376783, 3.39691121296312255949186788852, 4.33442413105742269514298860377, 4.88769856219563046032249857895, 5.99086930581468254593741185696, 7.13148746260481851165329497930, 8.303461522751645683616163526315, 8.359979664262145445390087772626, 9.477212860130141058202679779553