| L(s) = 1 | + (−2.66 − 0.402i)3-s + (−2.09 − 1.43i)5-s + (1.09 − 1.89i)7-s + (4.09 + 1.26i)9-s + (−0.694 − 3.04i)11-s + (0.257 − 3.43i)13-s + (5.02 + 4.66i)15-s + (−1.92 + 1.31i)17-s + (3.23 − 0.996i)19-s + (−3.68 + 4.61i)21-s + (−4.82 + 4.47i)23-s + (0.529 + 1.34i)25-s + (−3.13 − 1.51i)27-s + (−5.34 + 0.806i)29-s + (−0.778 + 1.98i)31-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.232i)3-s + (−0.938 − 0.639i)5-s + (0.413 − 0.715i)7-s + (1.36 + 0.421i)9-s + (−0.209 − 0.917i)11-s + (0.0713 − 0.951i)13-s + (1.29 + 1.20i)15-s + (−0.467 + 0.318i)17-s + (0.741 − 0.228i)19-s + (−0.803 + 1.00i)21-s + (−1.00 + 0.933i)23-s + (0.105 + 0.269i)25-s + (−0.603 − 0.290i)27-s + (−0.993 + 0.149i)29-s + (−0.139 + 0.356i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0539343 + 0.129255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0539343 + 0.129255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (6.03 - 2.55i)T \) |
| good | 3 | \( 1 + (2.66 + 0.402i)T + (2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (2.09 + 1.43i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 1.89i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.694 + 3.04i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.257 + 3.43i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (1.92 - 1.31i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.23 + 0.996i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (4.82 - 4.47i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (5.34 - 0.806i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (0.778 - 1.98i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (1.73 + 3.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.33 - 1.66i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (0.260 - 1.14i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.777 - 10.3i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-5.53 - 2.66i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.913 - 2.32i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (9.87 - 3.04i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (8.30 + 7.70i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.624 + 8.33i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-4.90 + 8.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.83 - 0.427i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 2.23i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (-0.939 - 4.11i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.40086732070854297220721488470, −9.013102658814905097243305398762, −7.87178721391184339574705850723, −7.45086125952658243605268433091, −6.14099582952456731704967866024, −5.43385837687216147915955921495, −4.55309567466421633207071232567, −3.53278181521140022369620274660, −1.18745355310808643147228748627, −0.10321322130043380316505516252,
2.07943461660372009211299657696, 3.84242647838753819309402373872, 4.73308689515081347745221306229, 5.56364973607961912705742230750, 6.62513758779362485951012792631, 7.24411423129636244090520136417, 8.317314438396662877841736607065, 9.542618975553228423979327565709, 10.36057518094168824814618410802, 11.23296648695432349844701507453
