| L(s) = 1 | + 1.41i·2-s + (1.10 + 2.78i)3-s − 2.00·4-s − 5.86i·5-s + (−3.94 + 1.56i)6-s − 9.12·7-s − 2.82i·8-s + (−6.54 + 6.17i)9-s + 8.28·10-s − 1.47i·11-s + (−2.21 − 5.57i)12-s + 3.43·13-s − 12.8i·14-s + (16.3 − 6.49i)15-s + 4.00·16-s − 28.5i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.369 + 0.929i)3-s − 0.500·4-s − 1.17i·5-s + (−0.657 + 0.261i)6-s − 1.30·7-s − 0.353i·8-s + (−0.727 + 0.686i)9-s + 0.828·10-s − 0.134i·11-s + (−0.184 − 0.464i)12-s + 0.264·13-s − 0.921i·14-s + (1.08 − 0.432i)15-s + 0.250·16-s − 1.67i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.525574 - 0.356715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.525574 - 0.356715i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-1.10 - 2.78i)T \) |
| 59 | \( 1 - 7.68iT \) |
| good | 5 | \( 1 + 5.86iT - 25T^{2} \) |
| 7 | \( 1 + 9.12T + 49T^{2} \) |
| 11 | \( 1 + 1.47iT - 121T^{2} \) |
| 13 | \( 1 - 3.43T + 169T^{2} \) |
| 17 | \( 1 + 28.5iT - 289T^{2} \) |
| 19 | \( 1 + 14.2T + 361T^{2} \) |
| 23 | \( 1 + 34.5iT - 529T^{2} \) |
| 29 | \( 1 - 18.1iT - 841T^{2} \) |
| 31 | \( 1 + 12.1T + 961T^{2} \) |
| 37 | \( 1 + 17.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 76.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.75T + 1.84e3T^{2} \) |
| 47 | \( 1 - 89.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 6.97iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 65.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 40.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 105. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 68.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 121.T + 9.40e3T^{2} \) |
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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
−10.80593962652373688478623744437, −9.844843991450450340699575146459, −8.973899315012092642462843182477, −8.703549565096569575017970496061, −7.29387428862155325436762027279, −6.09897681859806859643623055796, −5.05522880728948495694069946259, −4.21573824084873168994184429513, −2.92447483589562409386134575004, −0.26571317367113757674666089332,
1.81077453422311695815204064659, 3.06002115213269731811874677737, 3.73565637822163255996809880514, 5.98055396816712736145287532509, 6.56916987864030117549192751515, 7.61919213851357040801784935041, 8.700616393988606315268587665233, 9.745225794807808320200349069546, 10.53514407409939609353513133683, 11.44920685759233601340696706273
