| L(s) = 1 | + (1.82 − 2.38i)3-s − 7.37i·5-s + 2.64·7-s + (−2.35 − 8.68i)9-s − 2.61i·11-s − 6.35·13-s + (−17.5 − 13.4i)15-s + 12.1i·17-s + 10.2·19-s + (4.82 − 6.30i)21-s + 4.30i·23-s − 29.4·25-s + (−24.9 − 10.2i)27-s − 17.3i·29-s − 39.2·31-s + ⋯ |
| L(s) = 1 | + (0.607 − 0.794i)3-s − 1.47i·5-s + 0.377·7-s + (−0.261 − 0.965i)9-s − 0.237i·11-s − 0.488·13-s + (−1.17 − 0.896i)15-s + 0.714i·17-s + 0.538·19-s + (0.229 − 0.300i)21-s + 0.187i·23-s − 1.17·25-s + (−0.925 − 0.378i)27-s − 0.599i·29-s − 1.26·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.833665 - 1.68746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833665 - 1.68746i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.82 + 2.38i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 7.37iT - 25T^{2} \) |
| 11 | \( 1 + 2.61iT - 121T^{2} \) |
| 13 | \( 1 + 6.35T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 10.2T + 361T^{2} \) |
| 23 | \( 1 - 4.30iT - 529T^{2} \) |
| 29 | \( 1 + 17.3iT - 841T^{2} \) |
| 31 | \( 1 + 39.2T + 961T^{2} \) |
| 37 | \( 1 - 41.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 41.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 89.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.1T + 9.40e3T^{2} \) |
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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
−11.27999164036133624199097228638, −9.741632081212556290683511022675, −8.979816633591380998155419643329, −8.187037525808313048787365052044, −7.47909758234650719219270350101, −6.07538726898423923798627880906, −5.02448997658070927284224189854, −3.76870962771093580906676995185, −2.07121160080971664646636201333, −0.833994588421096671216963520256,
2.35743387254572341603819265958, 3.25673296809460727083687190440, 4.48127188121463098281824098756, 5.70621133444867641572537011631, 7.15245021020666141610955162648, 7.70704469118813221780996523076, 9.089538916202509050141811641322, 9.827181926166700370992617661242, 10.79754077445617304991951716153, 11.22591929824220692962922933213
