L(s) = 1 | + (−0.456 + 0.456i)3-s + (2.18 − 0.456i)5-s + (−2.79 + 2.79i)7-s + 2.58i·9-s + 4.37·11-s + (−1.73 − 1.73i)13-s + (−0.791 + 1.20i)15-s + (3 + 3i)17-s + 3.46i·19-s − 2.55i·21-s + (−0.791 − 0.791i)23-s + (4.58 − 1.99i)25-s + (−2.55 − 2.55i)27-s + 5.29·29-s + 1.58i·31-s + ⋯ |
L(s) = 1 | + (−0.263 + 0.263i)3-s + (0.978 − 0.204i)5-s + (−1.05 + 1.05i)7-s + 0.860i·9-s + 1.31·11-s + (−0.480 − 0.480i)13-s + (−0.204 + 0.312i)15-s + (0.727 + 0.727i)17-s + 0.794i·19-s − 0.556i·21-s + (−0.164 − 0.164i)23-s + (0.916 − 0.399i)25-s + (−0.490 − 0.490i)27-s + 0.982·29-s + 0.284i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13567 + 0.636091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13567 + 0.636091i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.18 + 0.456i)T \) |
good | 3 | \( 1 + (0.456 - 0.456i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.79 - 2.79i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (0.791 + 0.791i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 1.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 - 5.19i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + (-8.29 + 8.29i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.791 + 0.791i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.64 + 2.64i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 9.66iT - 61T^{2} \) |
| 67 | \( 1 + (8.29 + 8.29i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-0.582 + 0.582i)T - 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (4.83 - 4.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.16iT - 89T^{2} \) |
| 97 | \( 1 + (-0.582 - 0.582i)T + 97iT^{2} \) |
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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
−12.17245040808658280089626484929, −10.57002502485180181265324483685, −9.948551458618729811107302367865, −9.152423506336126777094516974725, −8.198625402990153660214975597408, −6.59554878058396533160034760266, −5.89505450297657342136389236909, −4.99797413585837926245795086769, −3.35546001902201512383705057784, −1.92554315144311601114371901541,
1.07028622373396869751093103667, 3.02196512345690803106487342514, 4.23458053509013375024023428511, 5.82203979151618119990072089480, 6.76234129169854604559939096921, 7.10974404488322538742096957428, 9.090071859657474306841660042453, 9.576355437309010828651576291404, 10.37526967242800635728219757041, 11.60537416940403308133765004850