| L(s) = 1 | + 1.03i·5-s + 0.378·11-s − 2·13-s + 3.86i·17-s − 1.46i·19-s + 5.27·23-s + 3.92·25-s − 3.48i·29-s − 2.53i·31-s + 2.92·37-s + 8.76i·41-s + 4i·43-s − 8.48·47-s − 7.07i·53-s + 0.392i·55-s + ⋯ |
| L(s) = 1 | + 0.462i·5-s + 0.114·11-s − 0.554·13-s + 0.937i·17-s − 0.335i·19-s + 1.10·23-s + 0.785·25-s − 0.647i·29-s − 0.455i·31-s + 0.481·37-s + 1.36i·41-s + 0.609i·43-s − 1.23·47-s − 0.971i·53-s + 0.0528i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790925749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790925749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 11 | \( 1 - 0.378T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 1.46iT - 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 3.48iT - 29T^{2} \) |
| 31 | \( 1 + 2.53iT - 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 - 8.76iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 8.53iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 1.03iT - 89T^{2} \) |
| 97 | \( 1 + 4.53T + 97T^{2} \) |
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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
−8.028528526368869731895034148946, −7.37944587415843749301159482734, −6.55447395156223811533937668349, −6.21276134791846831085502993922, −5.10804412284572960018460184126, −4.60786112365880416209534613973, −3.61029275144847038554149557052, −2.90269028603833005463153911059, −2.06526334142641229412013055848, −0.918088663860124931965254835671,
0.53416631046604951486466043464, 1.53917597194359588682873245510, 2.63568392018056124231595176882, 3.33719130648903421644563627017, 4.34721424599554989814654530556, 5.04811500737901156103280675597, 5.48354467918544878299904362006, 6.57514742973017969947013928642, 7.10642716414635758918435731275, 7.74442089146131604642372895658
