| L(s) = 1 | − i·2-s + (0.767 + 1.55i)3-s − 4-s − 5-s + (1.55 − 0.767i)6-s + i·8-s + (−1.82 + 2.38i)9-s + i·10-s + 2.30i·11-s + (−0.767 − 1.55i)12-s − 5.00i·13-s + (−0.767 − 1.55i)15-s + 16-s − 6.30·17-s + (2.38 + 1.82i)18-s + 7.55i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.443 + 0.896i)3-s − 0.5·4-s − 0.447·5-s + (0.633 − 0.313i)6-s + 0.353i·8-s + (−0.607 + 0.794i)9-s + 0.316i·10-s + 0.695i·11-s + (−0.221 − 0.448i)12-s − 1.38i·13-s + (−0.198 − 0.400i)15-s + 0.250·16-s − 1.52·17-s + (0.561 + 0.429i)18-s + 1.73i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2081604024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2081604024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.767 - 1.55i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2.30iT - 11T^{2} \) |
| 13 | \( 1 + 5.00iT - 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 7.55iT - 19T^{2} \) |
| 23 | \( 1 + 6.37iT - 23T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 - 4.38iT - 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.25iT - 53T^{2} \) |
| 59 | \( 1 + 9.93T + 59T^{2} \) |
| 61 | \( 1 + 5.98iT - 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 - 2.85iT - 71T^{2} \) |
| 73 | \( 1 - 3.68iT - 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + 5.88T + 89T^{2} \) |
| 97 | \( 1 - 4.61iT - 97T^{2} \) |
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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
−10.22771626553228500643260239979, −9.136357273717107286192746957470, −8.365708258136275204184880902923, −7.902505554733321007378839447957, −6.61167757256773350875250050639, −5.39960034690311792622430917660, −4.61562877824743531483672109011, −3.84406579577896019515911985813, −2.99542936773650487184821842002, −1.95023163907115415656344742406,
0.07420301348852106102760051344, 1.70373322795615010478690665938, 2.98297447635197373639459361381, 4.04909144672107741314296017308, 5.01056991738776800163593402145, 6.17485942266439875399625626301, 6.91753849806954861248524885980, 7.27141747214730816154542618073, 8.372755365630079966012674839594, 8.961899923701657536255834795209
