| L(s) = 1 | + (0.587 + 0.809i)2-s + (1.47 + 0.480i)3-s + (−0.309 + 0.951i)4-s + (2.21 − 0.333i)5-s + (0.480 + 1.47i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (−0.473 − 0.343i)9-s + (1.56 + 1.59i)10-s + (1.75 − 1.27i)11-s + (−0.913 + 1.25i)12-s + (0.0819 − 0.112i)13-s + (−0.809 + 0.587i)14-s + (3.42 + 0.569i)15-s + (−0.809 − 0.587i)16-s + (−4.61 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (0.853 + 0.277i)3-s + (−0.154 + 0.475i)4-s + (0.988 − 0.149i)5-s + (0.196 + 0.603i)6-s + 0.377i·7-s + (−0.336 + 0.109i)8-s + (−0.157 − 0.114i)9-s + (0.496 + 0.503i)10-s + (0.529 − 0.384i)11-s + (−0.263 + 0.362i)12-s + (0.0227 − 0.0312i)13-s + (−0.216 + 0.157i)14-s + (0.885 + 0.146i)15-s + (−0.202 − 0.146i)16-s + (−1.11 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98579 + 1.15453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98579 + 1.15453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-2.21 + 0.333i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (-1.47 - 0.480i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-1.75 + 1.27i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0819 + 0.112i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.61 - 1.49i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.0785 - 0.241i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.82 + 2.51i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.06 - 3.27i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.792 + 2.43i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.90 - 4.00i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.50 - 2.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.885iT - 43T^{2} \) |
| 47 | \( 1 + (-0.800 - 0.259i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.414 - 0.134i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.01 + 6.54i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.58 + 5.51i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (12.3 - 4.01i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.37 + 10.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.69 - 3.70i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.600 - 1.84i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-15.0 + 4.90i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.6 + 8.44i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.198 + 0.0644i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.79038750938541183997493991697, −10.61282896787296246768790243545, −9.375715459928080844465561938787, −8.915221273140802699704109784108, −8.078827245212072877221145713898, −6.58852042352098298895178390277, −5.93675051377189357034418285459, −4.69446277293811386955306965006, −3.43240752488683734495008618196, −2.20960167515821277537273070709,
1.77003641971832360046078865947, 2.70953616469599197968272633125, 4.03261517467551412469831516679, 5.32577259150705231384293658207, 6.47893683918609258982113284454, 7.50643190766889727673199511921, 8.874783842520045359752212616731, 9.412275347504981702385460811289, 10.45782746736746582684544783212, 11.27910013032814992273814218478
