| L(s) = 1 | + (−2.18 − 2.18i)3-s + (−1.79 − 1.79i)7-s + 6.58i·9-s − 0.913·11-s + (1.73 − 1.73i)13-s + (−3 + 3i)17-s − 3.46i·19-s + 7.84i·21-s + (−3.79 + 3.79i)23-s + (7.84 − 7.84i)27-s − 5.29·29-s + 7.58i·31-s + (1.99 + 1.99i)33-s + (5.19 + 5.19i)37-s − 7.58·39-s + ⋯ |
| L(s) = 1 | + (−1.26 − 1.26i)3-s + (−0.677 − 0.677i)7-s + 2.19i·9-s − 0.275·11-s + (0.480 − 0.480i)13-s + (−0.727 + 0.727i)17-s − 0.794i·19-s + 1.71i·21-s + (−0.790 + 0.790i)23-s + (1.50 − 1.50i)27-s − 0.982·29-s + 1.36i·31-s + (0.348 + 0.348i)33-s + (0.854 + 0.854i)37-s − 1.21·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4940064396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4940064396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (2.18 + 2.18i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.79 + 1.79i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.913T + 11T^{2} \) |
| 13 | \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (3 - 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (3.79 - 3.79i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 7.58iT - 31T^{2} \) |
| 37 | \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 + (0.361 + 0.361i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.79 - 3.79i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.64 - 2.64i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 6.20iT - 61T^{2} \) |
| 67 | \( 1 + (-0.361 + 0.361i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.41iT - 71T^{2} \) |
| 73 | \( 1 + (-8.58 - 8.58i)T + 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (3.10 + 3.10i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (-8.58 + 8.58i)T - 97iT^{2} \) |
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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
−9.569791233381164062457963483748, −8.384817446893144943037608793247, −7.61799822906326904149238828896, −6.90712759753193269852284666065, −6.29691239359637981620308065682, −5.62392059967302638218847699443, −4.65328132288850671927713452777, −3.42665209645905716682968171714, −2.02689933025414547026766138607, −0.882300295219546565703408251325,
0.29916255635019245667455245814, 2.37878349375421440626853787671, 3.74044684607450676197969025441, 4.33417175671797646946698620789, 5.35289710156263819853394618549, 6.01059496308033086352779418158, 6.51864038345380415362115986290, 7.77922749511758974761673726396, 9.044633927253720308098584549037, 9.406168796163658717781287993929
