| L(s) = 1 | + (−1.07 + 1.36i)3-s − 2.57·5-s + (−0.703 − 2.91i)9-s − 1.65i·11-s + 5.71i·13-s + (2.76 − 3.50i)15-s − 7.58·17-s − 2.99i·19-s + 0.287i·23-s + 1.65·25-s + (4.72 + 2.16i)27-s − 2.05i·29-s − 6.01i·31-s + (2.25 + 1.77i)33-s + 1.75·37-s + ⋯ |
| L(s) = 1 | + (−0.618 + 0.785i)3-s − 1.15·5-s + (−0.234 − 0.972i)9-s − 0.498i·11-s + 1.58i·13-s + (0.713 − 0.906i)15-s − 1.83·17-s − 0.686i·19-s + 0.0600i·23-s + 0.330·25-s + (0.908 + 0.417i)27-s − 0.382i·29-s − 1.08i·31-s + (0.391 + 0.308i)33-s + 0.288·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6912439578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6912439578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.07 - 1.36i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 11 | \( 1 + 1.65iT - 11T^{2} \) |
| 13 | \( 1 - 5.71iT - 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 2.99iT - 19T^{2} \) |
| 23 | \( 1 - 0.287iT - 23T^{2} \) |
| 29 | \( 1 + 2.05iT - 29T^{2} \) |
| 31 | \( 1 + 6.01iT - 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 + 0.373T + 47T^{2} \) |
| 53 | \( 1 - 7.77iT - 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 1.02iT - 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.81iT - 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 4.43iT - 97T^{2} \) |
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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
−9.062339260950715770139354880218, −8.429519947485930912790938733053, −7.35712333184829908692975497250, −6.63070409008295089490967606362, −5.96116898129659220519041942387, −4.64194995118998070443794960048, −4.35176632635231583934959057976, −3.58077158893194901718018843284, −2.28786065344744476169228286163, −0.45650115994225670298849661067,
0.62123307177165014629476828677, 2.02559777468044963754252426312, 3.14486943102818769530204725872, 4.22190731640258238760619451427, 5.02880934298936173857828181268, 5.87090254845432922401844545464, 6.84753953128961725913374080249, 7.31618259892954203903319224205, 8.219558610847785118944794880045, 8.508323774445490350751472483181
