| L(s) = 1 | + (−1.39 − 0.256i)2-s + (−1.23 − 1.21i)3-s + (1.86 + 0.714i)4-s + (1.46 + 1.74i)5-s + (1.40 + 2.00i)6-s + (−1.21 + 0.442i)7-s + (−2.41 − 1.47i)8-s + (0.0461 + 2.99i)9-s + (−1.58 − 2.80i)10-s + (−2.93 + 3.50i)11-s + (−1.43 − 3.15i)12-s + (4.95 − 0.873i)13-s + (1.80 − 0.303i)14-s + (0.313 − 3.93i)15-s + (2.98 + 2.66i)16-s + (0.909 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.983 − 0.181i)2-s + (−0.712 − 0.701i)3-s + (0.934 + 0.357i)4-s + (0.655 + 0.781i)5-s + (0.573 + 0.819i)6-s + (−0.459 + 0.167i)7-s + (−0.853 − 0.520i)8-s + (0.0153 + 0.999i)9-s + (−0.502 − 0.887i)10-s + (−0.885 + 1.05i)11-s + (−0.415 − 0.909i)12-s + (1.37 − 0.242i)13-s + (0.481 − 0.0810i)14-s + (0.0810 − 1.01i)15-s + (0.745 + 0.666i)16-s + (0.220 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.627596 + 0.177669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627596 + 0.177669i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.256i)T \) |
| 3 | \( 1 + (1.23 + 1.21i)T \) |
| good | 5 | \( 1 + (-1.46 - 1.74i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.21 - 0.442i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.93 - 3.50i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.95 + 0.873i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.909 - 1.57i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.49 - 3.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.835 - 0.304i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.50 - 0.265i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.86 + 0.679i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (8.19 - 4.73i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.859 - 4.87i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.62 - 1.93i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.64 + 2.78i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 3.70iT - 53T^{2} \) |
| 59 | \( 1 + (-3.63 - 4.32i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.89 + 13.4i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 1.42i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.59 + 7.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.72 - 9.90i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.39 - 7.90i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (13.3 + 2.34i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.65 + 6.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.14 - 1.79i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.28080122342267458772152703658, −11.25256090639946394599822657884, −10.35674750132315167112469589008, −9.838551407769321659951352332424, −8.298835059575366407216302918997, −7.34454222240643158839429156442, −6.44245247587532509158532027970, −5.58149552486521528647745491098, −3.07165738697689160456720154512, −1.64326529537699049284403475472,
0.885606298779145514126166616310, 3.28349139914440661949038928475, 5.31155242658518893597662655043, 5.87323346679151627366123423633, 7.15344655412243929943017737640, 8.728537532661565356702231528026, 9.198834598768490822074492447292, 10.26673857838873895622991343722, 10.99243881665364093013469522660, 11.86581554346089149146792306350
