L(s) = 1 | + 4.22i·2-s − 3·3-s − 9.85·4-s + 5.85i·5-s − 12.6i·6-s − 24.1i·7-s − 7.85i·8-s + 9·9-s − 24.7·10-s − 33.8i·11-s + 29.5·12-s + 101.·14-s − 17.5i·15-s − 45.6·16-s + 49.3·17-s + 38.0i·18-s + ⋯ |
L(s) = 1 | + 1.49i·2-s − 0.577·3-s − 1.23·4-s + 0.524i·5-s − 0.862i·6-s − 1.30i·7-s − 0.347i·8-s + 0.333·9-s − 0.783·10-s − 0.928i·11-s + 0.711·12-s + 1.94·14-s − 0.302i·15-s − 0.713·16-s + 0.704·17-s + 0.498i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.361385491\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.361385491\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 4.22iT - 8T^{2} \) |
| 5 | \( 1 - 5.85iT - 125T^{2} \) |
| 7 | \( 1 + 24.1iT - 343T^{2} \) |
| 11 | \( 1 + 33.8iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 49.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 6.29T + 1.21e4T^{2} \) |
| 29 | \( 1 - 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 307. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 76.0iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 514. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 67.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 25.2iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 588.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 895. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 968. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 480. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 16.6iT - 9.12e5T^{2} \) |
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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
−10.67929322533482222792785115889, −10.09996586529316291139306324629, −8.715917726808919628777279776749, −7.84386530226711883770577667684, −7.07911226129673112435065586566, −6.41035182610437586184043323154, −5.55282785305011076252829342194, −4.51301062447735934523132406122, −3.31389404118403436942887140501, −0.981693491481636556629495009090,
0.59132140320739781113644753036, 1.90802404368299994777597653302, 2.84067014431237230661366244718, 4.30673810389634406865984052269, 5.10121018583463566059219732400, 6.18419443726895411584425136935, 7.50478433272871216964318758094, 8.901223498808461405732354593463, 9.393087609835282213195942412346, 10.28779591101045225321249979592