| L(s) = 1 | + (−0.620 − 1.27i)2-s + (0.654 + 0.755i)3-s + (−1.23 + 1.57i)4-s + (0.514 + 3.57i)5-s + (0.554 − 1.30i)6-s + (0.174 − 0.382i)7-s + (2.76 + 0.586i)8-s + (−0.142 + 0.989i)9-s + (4.23 − 2.87i)10-s + (0.516 − 1.75i)11-s + (−1.99 + 0.102i)12-s + (−5.06 + 2.31i)13-s + (−0.594 + 0.0152i)14-s + (−2.36 + 2.73i)15-s + (−0.970 − 3.88i)16-s + (0.555 − 0.863i)17-s + ⋯ |
| L(s) = 1 | + (−0.438 − 0.898i)2-s + (0.378 + 0.436i)3-s + (−0.615 + 0.788i)4-s + (0.230 + 1.60i)5-s + (0.226 − 0.531i)6-s + (0.0660 − 0.144i)7-s + (0.978 + 0.207i)8-s + (−0.0474 + 0.329i)9-s + (1.33 − 0.908i)10-s + (0.155 − 0.530i)11-s + (−0.576 + 0.0295i)12-s + (−1.40 + 0.642i)13-s + (−0.158 + 0.00406i)14-s + (−0.611 + 0.705i)15-s + (−0.242 − 0.970i)16-s + (0.134 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.812135 + 0.666552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.812135 + 0.666552i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.620 + 1.27i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.11 + 2.47i)T \) |
| good | 5 | \( 1 + (-0.514 - 3.57i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.174 + 0.382i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.516 + 1.75i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (5.06 - 2.31i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.555 + 0.863i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.08 - 4.79i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.86 - 6.01i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.29 - 4.58i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.07 - 7.49i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.68 + 11.7i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.37 - 2.05i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 3.17iT - 47T^{2} \) |
| 53 | \( 1 + (-5.08 + 11.1i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.527 + 1.15i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-7.14 + 8.24i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.57 - 5.35i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.19 - 4.06i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (1.18 - 0.760i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.967 + 2.11i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-15.9 - 2.28i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (4.62 - 4.01i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (8.56 - 1.23i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.74530314018794471483108161478, −10.11560642719626468719013932677, −9.636384763721206519052216122453, −8.471658505793116483523604152131, −7.50571609493725655418715942061, −6.72432044785103574901611209124, −5.18792581382723010504258607137, −3.82974455262924023011760550661, −3.04364265257926551951047620929, −2.05308300792522992283114304157,
0.66665671206797121218430684228, 2.12639330277774153873872088178, 4.30651999292521261518450671309, 5.14207308687950508542747208414, 5.93727211396500448527373828743, 7.31532678481818139421661756389, 7.86798825934968700839080905988, 8.746527535981213072862335139147, 9.557644586604255391883854409358, 9.962633469317601557747810081874
