| L(s) = 1 | + (−2.06 − 1.85i)2-s + (−0.994 + 0.104i)3-s + (0.595 + 5.66i)4-s + (−2.11 − 0.713i)5-s + (2.24 + 1.63i)6-s + (−1.97 − 1.75i)7-s + (6.02 − 8.28i)8-s + (0.978 − 0.207i)9-s + (3.04 + 5.40i)10-s + (−5.77 − 1.22i)11-s + (−1.18 − 5.56i)12-s + (−1.69 − 0.550i)13-s + (0.815 + 7.29i)14-s + (2.18 + 0.487i)15-s + (−16.6 + 3.54i)16-s + (−0.431 − 0.970i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 − 1.31i)2-s + (−0.574 + 0.0603i)3-s + (0.297 + 2.83i)4-s + (−0.947 − 0.318i)5-s + (0.916 + 0.665i)6-s + (−0.747 − 0.664i)7-s + (2.12 − 2.93i)8-s + (0.326 − 0.0693i)9-s + (0.962 + 1.70i)10-s + (−1.74 − 0.370i)11-s + (−0.341 − 1.60i)12-s + (−0.470 − 0.152i)13-s + (0.218 + 1.94i)14-s + (0.563 + 0.125i)15-s + (−4.16 + 0.885i)16-s + (−0.104 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.160559 - 0.0180694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160559 - 0.0180694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (2.11 + 0.713i)T \) |
| 7 | \( 1 + (1.97 + 1.75i)T \) |
| good | 2 | \( 1 + (2.06 + 1.85i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (5.77 + 1.22i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.69 + 0.550i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.431 + 0.970i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.320 - 3.04i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.664 - 0.598i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-5.67 + 4.12i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.21 + 0.538i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.798 + 3.75i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (2.22 - 6.83i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.18iT - 43T^{2} \) |
| 47 | \( 1 + (3.01 - 6.76i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.90 + 0.305i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.739 - 0.821i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-8.98 + 9.97i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.66 + 5.98i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-2.66 + 1.93i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.43 + 6.74i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-14.3 - 6.38i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.236 + 0.325i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (9.16 - 10.1i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-4.20 - 5.79i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.72112748807422056123023141898, −10.13643449430701395290007015058, −9.371447260847086692749163989673, −7.956788436373316074106618748610, −7.903953626332265028822285147907, −6.68914955424089339868071670937, −4.79361455579450861998124134456, −3.64867523640876970880629901773, −2.68163734494056993013213584715, −0.73420891342561945715740180310,
0.26250966449618872561195361678, 2.51414424079815607581954233563, 4.84731287566403045794657424950, 5.56411324571306571349633092165, 6.79397092692751844344069356871, 7.14806498257991803716687311446, 8.204201081716539640289429594564, 8.819142840761586294130338041693, 10.10321541486994632953542208596, 10.39904683430686702949517092171
