| L(s) = 1 | + (2.13 + 0.245i)2-s + (−1.59 − 1.16i)3-s + (2.56 + 0.596i)4-s + (−0.564 + 0.825i)5-s + (−3.13 − 2.87i)6-s + (−2.87 − 2.95i)7-s + (1.28 + 0.459i)8-s + (0.279 + 0.860i)9-s + (−1.40 + 1.62i)10-s + (−3.08 + 1.21i)11-s + (−3.40 − 3.93i)12-s + (2.25 − 3.73i)13-s + (−5.42 − 7.03i)14-s + (1.86 − 0.664i)15-s + (−2.08 − 1.02i)16-s + (0.584 − 0.478i)17-s + ⋯ |
| L(s) = 1 | + (1.51 + 0.173i)2-s + (−0.922 − 0.670i)3-s + (1.28 + 0.298i)4-s + (−0.252 + 0.369i)5-s + (−1.27 − 1.17i)6-s + (−1.08 − 1.11i)7-s + (0.455 + 0.162i)8-s + (0.0932 + 0.286i)9-s + (−0.445 + 0.514i)10-s + (−0.930 + 0.365i)11-s + (−0.984 − 1.13i)12-s + (0.624 − 1.03i)13-s + (−1.45 − 1.88i)14-s + (0.480 − 0.171i)15-s + (−0.521 − 0.256i)16-s + (0.141 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563163 - 1.19830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563163 - 1.19830i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (3.08 - 1.21i)T \) |
| good | 2 | \( 1 + (-2.13 - 0.245i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (1.59 + 1.16i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (2.87 + 2.95i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-2.25 + 3.73i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.584 + 0.478i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (0.538 + 0.0307i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.613 + 4.26i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-0.886 + 2.27i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (-4.74 - 2.00i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.267 - 3.11i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (8.80 - 4.96i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-6.67 + 1.95i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.180 + 0.892i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-6.87 + 3.38i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (9.92 + 5.60i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (-13.9 + 1.60i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (-4.83 + 10.5i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (2.37 + 9.05i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-7.30 - 13.8i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.135 - 4.74i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (8.48 + 1.46i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (10.7 + 6.90i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.95 + 4.31i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.62502169917611147175728680611, −9.913855603580246778074568051096, −8.146092614550506488125503797922, −7.04067728013754877169839805427, −6.60165434427931877715176684711, −5.81496535181392318645664135762, −4.85158834266699195564616092071, −3.69357732508999743342170408566, −2.86002563383433351315805547397, −0.47492768377844020272035434245,
2.47691594873618837280718407601, 3.59449182273153126392513468757, 4.52349748354095020635119952775, 5.57078648985945012247337241569, 5.79280965373430661609568652408, 6.85111433872147652220009556509, 8.483778170951379563487730782806, 9.358884343534380827207645097899, 10.42181569388319117051060374582, 11.32959006665096792236724994535
