| L(s) = 1 | + (1.89 − 2.19i)3-s + (−0.556 + 3.86i)5-s + (−1.06 − 2.32i)7-s + (−0.768 − 5.34i)9-s + (−2.52 + 0.740i)11-s + (−0.904 + 1.98i)13-s + (7.41 + 8.56i)15-s + (−0.0617 + 0.0396i)17-s + (0.698 + 0.448i)19-s + (−7.11 − 2.08i)21-s + (1.49 + 4.55i)23-s + (−9.85 − 2.89i)25-s + (−5.85 − 3.76i)27-s + (6.31 − 4.06i)29-s + (−4.12 − 4.75i)31-s + ⋯ |
| L(s) = 1 | + (1.09 − 1.26i)3-s + (−0.248 + 1.73i)5-s + (−0.401 − 0.879i)7-s + (−0.256 − 1.78i)9-s + (−0.760 + 0.223i)11-s + (−0.250 + 0.549i)13-s + (1.91 + 2.21i)15-s + (−0.0149 + 0.00962i)17-s + (0.160 + 0.102i)19-s + (−1.55 − 0.456i)21-s + (0.312 + 0.949i)23-s + (−1.97 − 0.578i)25-s + (−1.12 − 0.723i)27-s + (1.17 − 0.754i)29-s + (−0.740 − 0.854i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16399 - 0.306995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16399 - 0.306995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-1.49 - 4.55i)T \) |
| good | 3 | \( 1 + (-1.89 + 2.19i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.556 - 3.86i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.06 + 2.32i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.52 - 0.740i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.904 - 1.98i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.0617 - 0.0396i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 0.448i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.31 + 4.06i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.12 + 4.75i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.834 - 5.80i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.336 + 2.34i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.43 + 6.27i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + (-0.347 - 0.761i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.679 - 1.48i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.80 + 9.00i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.65 - 0.486i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.30 + 0.677i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.00 - 5.78i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.03 + 6.65i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.102 - 0.711i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.113 + 0.131i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (2.24 - 15.6i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−13.86652144825101062976829926149, −13.33971349160696780148049528502, −11.97550627019699408590801388159, −10.73565344286001388560380260501, −9.632989293819889628230476417178, −7.898011859329124275024836811409, −7.26768279940361326719009294049, −6.48833540363485734680428039114, −3.57041507171523526268934603683, −2.42830682292154710101942505897,
2.91650208968170250628165228900, 4.53053243782054121900328843158, 5.40533439964215059638747484880, 8.053094486670650645565230345928, 8.790597355927062754402848924054, 9.435137649634451867763163013449, 10.63191574936303038310175827432, 12.34614513031897125726141428718, 13.01575991722383143456136535991, 14.31687022823071820803189326712
