| L(s) = 1 | + (0.210 − 1.46i)3-s + (0.926 − 0.272i)5-s + (−1.14 − 1.32i)7-s + (0.785 + 0.230i)9-s + (0.436 + 0.280i)11-s + (−2.04 + 2.35i)13-s + (−0.203 − 1.41i)15-s + (1.39 + 3.05i)17-s + (−2.90 + 6.36i)19-s + (−2.17 + 1.40i)21-s + (−1.35 − 4.60i)23-s + (−3.42 + 2.19i)25-s + (2.34 − 5.13i)27-s + (1.01 + 2.21i)29-s + (−0.253 − 1.76i)31-s + ⋯ |
| L(s) = 1 | + (0.121 − 0.843i)3-s + (0.414 − 0.121i)5-s + (−0.434 − 0.501i)7-s + (0.261 + 0.0768i)9-s + (0.131 + 0.0845i)11-s + (−0.566 + 0.654i)13-s + (−0.0524 − 0.364i)15-s + (0.338 + 0.740i)17-s + (−0.666 + 1.46i)19-s + (−0.475 + 0.305i)21-s + (−0.282 − 0.959i)23-s + (−0.684 + 0.439i)25-s + (0.450 − 0.987i)27-s + (0.188 + 0.412i)29-s + (−0.0454 − 0.316i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.974050 - 0.385623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974050 - 0.385623i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (1.35 + 4.60i)T \) |
| good | 3 | \( 1 + (-0.210 + 1.46i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.926 + 0.272i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.14 + 1.32i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.436 - 0.280i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.04 - 2.35i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.39 - 3.05i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.90 - 6.36i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.01 - 2.21i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.253 + 1.76i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.58 + 1.34i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-12.0 + 3.54i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.236 - 1.64i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 5.46T + 47T^{2} \) |
| 53 | \( 1 + (8.08 + 9.32i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.434 - 0.500i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.685 + 4.76i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (9.51 - 6.11i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (4.30 - 2.76i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.40 + 11.8i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (9.56 - 11.0i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.91 - 0.854i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.78 - 12.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.645 - 0.189i)T + (81.6 - 52.4i)T^{2} \) |
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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
−13.87746279969656257270825461503, −12.79064336001435995505304768343, −12.23733338297074728087565847737, −10.57663904045988798365334061599, −9.655645674845707814589151454928, −8.179486869148536406465343017569, −7.07764940851475640223032654103, −6.01656916805630417788144171389, −4.10055755733027104876728142176, −1.88651142605817688110252411733,
2.87271157771041116109145278792, 4.55649725117800984101633133659, 5.90621966047291652155226060729, 7.38914735333399448557058400501, 9.073880864857998544431910803267, 9.738496449049352445334333549252, 10.76083190376504744449616307848, 12.09424949287418530683732839216, 13.17792279286181374605871329143, 14.29643599728722020389167437357
