| 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
−11.91883185416965697900150335017, −10.27755945043939667531686957622, −9.359780733029339337308336702454, −8.708544481251540904870899282794, −7.49311172913828582315486874255, −6.77236414438607065854888378795, −5.05638090604847848254266893531, −3.99222322659000759568412680723, −2.70620218928844288017992089977, −1.50405274812456694931217389487,
2.87222306483041117749555418816, 3.53208190298536566895395089952, 4.54347228576911265021242136493, 6.12720673157605482870309132701, 7.10141446548537294679801714800, 8.095176109186281176954318968475, 9.122467939881591191205416209730, 9.650053538781753548007827471256, 11.05161007450194924287224846738, 11.74954520826569983127445187237
