| L(s) = 1 | + (−0.959 − 0.281i)3-s + (2.00 − 1.29i)5-s + (0.340 − 2.36i)7-s + (0.841 + 0.540i)9-s + (−0.459 − 1.00i)11-s + (0.117 + 0.817i)13-s + (−2.29 + 0.672i)15-s + (−0.0178 + 0.0206i)17-s + (−2.16 − 2.49i)19-s + (−0.993 + 2.17i)21-s + (−2.91 − 3.81i)23-s + (0.291 − 0.637i)25-s + (−0.654 − 0.755i)27-s + (5.03 − 5.80i)29-s + (2.85 − 0.837i)31-s + ⋯ |
| L(s) = 1 | + (−0.553 − 0.162i)3-s + (0.898 − 0.577i)5-s + (0.128 − 0.895i)7-s + (0.280 + 0.180i)9-s + (−0.138 − 0.303i)11-s + (0.0325 + 0.226i)13-s + (−0.591 + 0.173i)15-s + (−0.00433 + 0.00499i)17-s + (−0.496 − 0.572i)19-s + (−0.216 + 0.474i)21-s + (−0.606 − 0.794i)23-s + (0.0582 − 0.127i)25-s + (−0.126 − 0.145i)27-s + (0.934 − 1.07i)29-s + (0.512 − 0.150i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.974324 - 0.876086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974324 - 0.876086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (2.91 + 3.81i)T \) |
| good | 5 | \( 1 + (-2.00 + 1.29i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.340 + 2.36i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.459 + 1.00i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.117 - 0.817i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.0178 - 0.0206i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.16 + 2.49i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-5.03 + 5.80i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.85 + 0.837i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.444 + 0.285i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.728 + 0.468i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (0.841 + 0.247i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 + (-1.44 + 10.0i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.787 + 5.47i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (3.25 - 0.956i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.92 - 10.7i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.565 + 1.23i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (0.761 + 0.878i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.04 - 14.2i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (3.18 + 2.04i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-7.92 - 2.32i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (9.52 - 6.12i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.48333453835004753499352519428, −9.901853563753543556863437416768, −8.844723428082715027336633683752, −7.909614188507808642700926572462, −6.76725160151512406427613347410, −6.02860451789956209023253064827, −4.97893068710564356666506727435, −4.09106590850896827806273829453, −2.28528907589826096488194336190, −0.835385726778916553157743010981,
1.81648675265524355108583672496, 3.00493689618842425291217315260, 4.55119134032410856888938883780, 5.68386664042138353417439119947, 6.15551657527005154483593156456, 7.25131720752728341022034183233, 8.438235407351429591814619653757, 9.393079549335392443652565228213, 10.24224519199722597146064959788, 10.77454289809378580157637495851
