| L(s) = 1 | + (−0.826 + 0.563i)2-s + (2.11 − 1.96i)3-s + (0.365 − 0.930i)4-s + (−0.596 − 0.184i)5-s + (−0.643 + 2.81i)6-s + (−2.13 − 1.55i)7-s + (0.222 + 0.974i)8-s + (0.400 − 5.33i)9-s + (0.596 − 0.184i)10-s + (0.449 + 5.99i)11-s + (−1.05 − 2.69i)12-s + (2.76 + 1.33i)13-s + (2.64 + 0.0805i)14-s + (−1.62 + 0.783i)15-s + (−0.733 − 0.680i)16-s + (4.51 + 0.680i)17-s + ⋯ |
| L(s) = 1 | + (−0.584 + 0.398i)2-s + (1.22 − 1.13i)3-s + (0.182 − 0.465i)4-s + (−0.266 − 0.0823i)5-s + (−0.262 + 1.15i)6-s + (−0.808 − 0.588i)7-s + (0.0786 + 0.344i)8-s + (0.133 − 1.77i)9-s + (0.188 − 0.0582i)10-s + (0.135 + 1.80i)11-s + (−0.304 − 0.776i)12-s + (0.767 + 0.369i)13-s + (0.706 + 0.0215i)14-s + (−0.419 + 0.202i)15-s + (−0.183 − 0.170i)16-s + (1.09 + 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.960758 - 0.294671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960758 - 0.294671i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 7 | \( 1 + (2.13 + 1.55i)T \) |
| good | 3 | \( 1 + (-2.11 + 1.96i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (0.596 + 0.184i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.449 - 5.99i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 1.33i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-4.51 - 0.680i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (1.17 - 2.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.00 - 0.603i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (1.66 - 2.08i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (1.94 + 3.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.562 - 1.43i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.156 - 0.687i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.34 + 10.2i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (7.63 - 5.20i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.30 + 5.88i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (1.71 - 0.529i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (1.42 + 3.63i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.65 + 9.60i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 7.63i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (4.68 - 8.10i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.36 + 3.06i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.531 + 7.09i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−13.92997018127787140575101998628, −12.87432372723941908410921271226, −12.07405853666849602925241745017, −10.13726242342792036703466587337, −9.319193152197473627760232931317, −8.017344547926627032394866193352, −7.35887112111507550998847118419, −6.35946038369244157806233597888, −3.79970833619456491519413517479, −1.82854947013467717533090317277,
3.00527237706036160122472897726, 3.69632383036720937259094752278, 5.88380188362673842901445782251, 7.974370501580096538949080706872, 8.739132381189231903210541686525, 9.556964840536417231997570562227, 10.55564618900984823849259660325, 11.60673127428867994475724275920, 13.14825009182548409843036137031, 14.06143149230161291236332821340
