| L(s) = 1 | + (−2.53 + 1.60i)3-s + (−4.69 + 8.13i)7-s + (3.82 − 8.14i)9-s + (15.4 + 8.90i)11-s + (6.92 + 11.9i)13-s − 30.7i·17-s + 28.5·19-s + (−1.19 − 28.1i)21-s + (−10.6 + 6.16i)23-s + (3.42 + 26.7i)27-s + (−5.02 − 2.90i)29-s + (−3.25 − 5.63i)31-s + (−53.3 + 2.26i)33-s + 66.5·37-s + (−36.8 − 19.2i)39-s + ⋯ |
| L(s) = 1 | + (−0.844 + 0.536i)3-s + (−0.670 + 1.16i)7-s + (0.424 − 0.905i)9-s + (1.40 + 0.809i)11-s + (0.532 + 0.922i)13-s − 1.80i·17-s + 1.50·19-s + (−0.0568 − 1.34i)21-s + (−0.464 + 0.267i)23-s + (0.126 + 0.991i)27-s + (−0.173 − 0.100i)29-s + (−0.104 − 0.181i)31-s + (−1.61 + 0.0686i)33-s + 1.79·37-s + (−0.943 − 0.492i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.303845893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303845893\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.53 - 1.60i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (4.69 - 8.13i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.4 - 8.90i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-6.92 - 11.9i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 30.7iT - 289T^{2} \) |
| 19 | \( 1 - 28.5T + 361T^{2} \) |
| 23 | \( 1 + (10.6 - 6.16i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (5.02 + 2.90i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (3.25 + 5.63i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 66.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (33.0 - 19.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (27.5 - 47.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-14.2 - 8.23i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 69.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-91.2 + 52.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.5 - 58.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (22.9 + 39.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (47.3 - 81.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-13.4 - 7.73i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 52.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (38.1 - 66.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.720064987319238184893597535489, −9.578052103119014896552010619673, −8.965474820892919911748426720609, −7.38096905768989266618058136389, −6.57580453994969792175893634513, −5.88793923712427698280473726456, −4.92348905340871648494114602818, −4.01396873246068418901675218760, −2.84499441634053913348742364727, −1.22904095531808603704517631899,
0.58105410838605969762331896550, 1.44599535795504893080039714513, 3.43159936109038234442394833849, 4.06959264481281811617026097512, 5.55834115189466324431278221991, 6.21700589115242612287200577135, 6.91870655293434396529388490257, 7.82635999389487193550024918060, 8.690205209915048663479068555725, 9.977388281427767715258274720648
