L(s) = 1 | + (−1.65 + 0.524i)3-s + (−1.01 + 1.75i)5-s + (−4.09 + 2.36i)7-s + (2.44 − 1.73i)9-s + (−2.93 + 1.69i)11-s + (−5.51 − 3.18i)13-s + (0.751 − 3.42i)15-s − 4.87i·17-s + 1.48·19-s + (5.51 − 6.04i)21-s + (0.751 − 1.30i)23-s + (0.449 + 0.778i)25-s + (−3.13 + 4.14i)27-s + (3.49 + 6.04i)29-s + (5.93 + 3.42i)31-s + ⋯ |
L(s) = 1 | + (−0.953 + 0.302i)3-s + (−0.452 + 0.784i)5-s + (−1.54 + 0.893i)7-s + (0.816 − 0.577i)9-s + (−0.883 + 0.510i)11-s + (−1.53 − 0.883i)13-s + (0.193 − 0.884i)15-s − 1.18i·17-s + 0.340·19-s + (1.20 − 1.32i)21-s + (0.156 − 0.271i)23-s + (0.0898 + 0.155i)25-s + (−0.603 + 0.797i)27-s + (0.648 + 1.12i)29-s + (1.06 + 0.615i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2919032113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2919032113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 - 0.524i)T \) |
good | 5 | \( 1 + (1.01 - 1.75i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (4.09 - 2.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.51 + 3.18i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.87iT - 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + (-0.751 + 1.30i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.49 - 6.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.93 - 3.42i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.86iT - 37T^{2} \) |
| 41 | \( 1 + (-4.62 - 2.66i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 4.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.09 + 7.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.96T + 53T^{2} \) |
| 59 | \( 1 + (7.88 + 4.55i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.51 + 3.18i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.48 + 9.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.68T + 71T^{2} \) |
| 73 | \( 1 - 0.449T + 73T^{2} \) |
| 79 | \( 1 + (5.93 - 3.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 3.50i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.142iT - 89T^{2} \) |
| 97 | \( 1 + (-0.724 - 1.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.900741389562299950894807189969, −9.238914123023602007968674882825, −7.79705839901928598603072094049, −6.99056305700415686436615562909, −6.47357771110597885500559669698, −5.31000699635562838190384298264, −4.82866765849398668166562754407, −3.11975911516867419166084217044, −2.79379894633875621441319675391, −0.21084567416037962076272138735,
0.78035815004963143893333396067, 2.57697219414131243507273408301, 4.03615742711324605361914285235, 4.64694776020012773559721034399, 5.79583474237896992450077065728, 6.51171569419134550493201549042, 7.37643287221596746008380512581, 8.013058449081410164650763625673, 9.285827532108444149737494246695, 10.04484859228185204693394078379