| L(s) = 1 | + (0.654 + 0.755i)2-s + (2.09 + 1.34i)3-s + (−0.142 + 0.989i)4-s + (−0.762 + 1.66i)5-s + (0.353 + 2.46i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (1.32 + 2.89i)9-s + (−1.76 + 0.517i)10-s + (4.20 − 4.85i)11-s + (−1.62 + 1.87i)12-s + (−4.06 + 1.19i)13-s + (−0.415 − 0.909i)14-s + (−3.84 + 2.46i)15-s + (−0.959 − 0.281i)16-s + (0.235 + 1.63i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (1.20 + 0.776i)3-s + (−0.0711 + 0.494i)4-s + (−0.341 + 0.746i)5-s + (0.144 + 1.00i)6-s + (−0.362 − 0.106i)7-s + (−0.297 + 0.191i)8-s + (0.441 + 0.965i)9-s + (−0.556 + 0.163i)10-s + (1.26 − 1.46i)11-s + (−0.470 + 0.542i)12-s + (−1.12 + 0.330i)13-s + (−0.111 − 0.243i)14-s + (−0.991 + 0.637i)15-s + (−0.239 − 0.0704i)16-s + (0.0570 + 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40781 + 1.66247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40781 + 1.66247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-2.63 - 4.00i)T \) |
| good | 3 | \( 1 + (-2.09 - 1.34i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (0.762 - 1.66i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-4.20 + 4.85i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.06 - 1.19i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.235 - 1.63i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.662 + 4.60i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (1.02 + 7.09i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.894 + 0.574i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.165 + 0.361i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (2.68 - 5.87i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.34 - 4.07i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 + (4.95 + 1.45i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (12.7 - 3.75i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-12.0 + 7.75i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.679 - 0.783i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (8.01 + 9.24i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.415 - 2.88i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.26 - 0.666i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.35 + 2.96i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (8.37 + 5.38i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.297 + 0.651i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.74954520826569983127445187237, −11.05161007450194924287224846738, −9.650053538781753548007827471256, −9.122467939881591191205416209730, −8.095176109186281176954318968475, −7.10141446548537294679801714800, −6.12720673157605482870309132701, −4.54347228576911265021242136493, −3.53208190298536566895395089952, −2.87222306483041117749555418816,
1.50405274812456694931217389487, 2.70620218928844288017992089977, 3.99222322659000759568412680723, 5.05638090604847848254266893531, 6.77236414438607065854888378795, 7.49311172913828582315486874255, 8.708544481251540904870899282794, 9.359780733029339337308336702454, 10.27755945043939667531686957622, 11.91883185416965697900150335017
