| L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (2.30 − 1.81i)5-s + (−0.841 − 0.540i)6-s + (−2.63 + 0.242i)7-s + (0.654 − 0.755i)8-s + (0.928 − 0.371i)9-s + (−2.92 − 0.279i)10-s + (2.65 − 2.53i)11-s + (0.235 + 0.971i)12-s + (1.50 + 3.30i)13-s + (2.07 + 1.64i)14-s + (1.92 − 2.21i)15-s + (−0.995 + 0.0950i)16-s + (1.58 − 0.819i)17-s + ⋯ |
| L(s) = 1 | + (−0.511 − 0.487i)2-s + (0.566 − 0.109i)3-s + (0.0237 + 0.499i)4-s + (1.03 − 0.811i)5-s + (−0.343 − 0.220i)6-s + (−0.995 + 0.0917i)7-s + (0.231 − 0.267i)8-s + (0.309 − 0.123i)9-s + (−0.924 − 0.0882i)10-s + (0.800 − 0.763i)11-s + (0.0680 + 0.280i)12-s + (0.418 + 0.916i)13-s + (0.554 + 0.438i)14-s + (0.496 − 0.572i)15-s + (−0.248 + 0.0237i)16-s + (0.385 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31269 - 1.10648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31269 - 1.10648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.723 + 0.690i)T \) |
| 3 | \( 1 + (-0.981 + 0.189i)T \) |
| 7 | \( 1 + (2.63 - 0.242i)T \) |
| 23 | \( 1 + (-2.69 + 3.96i)T \) |
| good | 5 | \( 1 + (-2.30 + 1.81i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (-2.65 + 2.53i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.50 - 3.30i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 0.819i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-2.11 - 1.08i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (6.06 + 3.89i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.452 + 1.30i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (-4.97 + 1.99i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (-0.177 - 1.23i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.52 + 8.68i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 2.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.61 - 10.6i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (-3.59 - 0.343i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (7.12 + 1.37i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (-1.54 + 6.38i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (7.41 - 2.17i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.0129 + 0.272i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (3.21 - 4.50i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (-1.78 + 12.3i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (0.505 - 1.45i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-2.03 - 14.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−9.542084024895425343181493100556, −9.130718269984375164314732111328, −8.641820719932940171657082829376, −7.40991156172949046088443767896, −6.40530537514138502913916552315, −5.67690941707107683029705414542, −4.21924111246522581094071335849, −3.28188333546261767373674679255, −2.11491486162748011111779050442, −0.992902061653938094843650443603,
1.48901002973880840174230181930, 2.81447003447481437842688755771, 3.67856347797694413898787710788, 5.27570817735374426410788710519, 6.13475555077446965872185635941, 6.89451960351726086970026353288, 7.52040454168859146617357227171, 8.710209878132132155706772381384, 9.550918092231862128933075012295, 9.885181798590546701755044664992
