| L(s) = 1 | + (−0.401 + 1.23i)2-s + (0.809 + 0.587i)3-s + (0.253 + 0.183i)4-s + (−0.860 + 2.06i)5-s + (−1.05 + 0.763i)6-s − 7-s + (−2.43 + 1.76i)8-s + (0.309 + 0.951i)9-s + (−2.20 − 1.89i)10-s + (−0.475 + 1.46i)11-s + (0.0967 + 0.297i)12-s + (−0.626 − 1.92i)13-s + (0.401 − 1.23i)14-s + (−1.90 + 1.16i)15-s + (−1.01 − 3.11i)16-s + (0.317 − 0.230i)17-s + ⋯ |
| L(s) = 1 | + (−0.283 + 0.873i)2-s + (0.467 + 0.339i)3-s + (0.126 + 0.0919i)4-s + (−0.384 + 0.923i)5-s + (−0.428 + 0.311i)6-s − 0.377·7-s + (−0.859 + 0.624i)8-s + (0.103 + 0.317i)9-s + (−0.697 − 0.598i)10-s + (−0.143 + 0.441i)11-s + (0.0279 + 0.0859i)12-s + (−0.173 − 0.534i)13-s + (0.107 − 0.330i)14-s + (−0.492 + 0.300i)15-s + (−0.253 − 0.778i)16-s + (0.0769 − 0.0558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0519983 - 1.15882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0519983 - 1.15882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.860 - 2.06i)T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + (0.401 - 1.23i)T + (-1.61 - 1.17i)T^{2} \) |
| 11 | \( 1 + (0.475 - 1.46i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.626 + 1.92i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.317 + 0.230i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.666 - 0.484i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.225 - 0.693i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.95 - 2.87i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.01 + 3.64i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.09 - 9.54i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.46 - 7.58i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 + (-2.52 - 1.83i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (11.1 + 8.11i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.688 - 2.12i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.0109 + 0.0337i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (1.49 - 1.08i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.42 - 6.12i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.96 + 6.05i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.10 - 2.98i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.14 - 1.56i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.601 - 1.85i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-15.5 - 11.2i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.24623191397650674726298001169, −10.22116406836465478769795231859, −9.546325835407211551608914119862, −8.255309785236490400635292381674, −7.82920947667666768260560998165, −6.82660366880090118822135548373, −6.17485150464929671721680047213, −4.78604288397744679987010232116, −3.35266176547507492421793540830, −2.58230037104678042860271777993,
0.68450691099234700885814047042, 2.05984126612919388726829266625, 3.24870115464849299204534750231, 4.36685270307639136908083589296, 5.80309959516660744926424231498, 6.80229087894283020558548947942, 7.932067470352859862911320177497, 8.882837095934762110118467622136, 9.441328517281515707702814550662, 10.42603074041146850638573803117
