L(s) = 1 | − 0.114i·5-s + 7-s + 0.412i·11-s + 1.73i·13-s + 2.50·17-s + 6.85i·19-s − 4.42·23-s + 4.98·25-s − 1.85i·29-s − 5.60·31-s − 0.114i·35-s − 4.39i·37-s − 2.39·41-s + 4.35i·43-s + 7.23·47-s + ⋯ |
L(s) = 1 | − 0.0512i·5-s + 0.377·7-s + 0.124i·11-s + 0.481i·13-s + 0.608·17-s + 1.57i·19-s − 0.922·23-s + 0.997·25-s − 0.344i·29-s − 1.00·31-s − 0.0193i·35-s − 0.721i·37-s − 0.374·41-s + 0.664i·43-s + 1.05·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.462261034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.462261034\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + 0.114iT - 5T^{2} \) |
| 11 | \( 1 - 0.412iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 - 6.85iT - 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 + 1.85iT - 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 + 4.39iT - 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 4.35iT - 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 4.25iT - 59T^{2} \) |
| 61 | \( 1 - 7.35iT - 61T^{2} \) |
| 67 | \( 1 - 6.25iT - 67T^{2} \) |
| 71 | \( 1 - 0.608T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 + 4.88iT - 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−8.267406995634446613572145738774, −7.54215294995282156659198429753, −7.02798799705246632092323335909, −5.85645170362532475654374870236, −5.69765532067549853730618519147, −4.51189541686781446744369742865, −3.99446658858580949974860336730, −3.06107389508538299237720090497, −2.00806395052403941760453756993, −1.21538550115896424608599754056,
0.38096039076953683893972454969, 1.53219022979678868185813734659, 2.59014465134489468388595444924, 3.33404579739376068442276156815, 4.25134174198101010050132318619, 5.09606146023762239277397141598, 5.57762490179219351833578652699, 6.56881386680323976712724086733, 7.16317484044592989789425886083, 7.85932342346967763186355694393