L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s + (−1.73 + i)7-s + 0.999i·8-s + (0.5 − 0.866i)10-s + (−5.59 − 3.23i)11-s + (1 − 3.46i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (6.46 − 3.73i)19-s + (0.866 − 0.499i)20-s + (−3.23 − 5.59i)22-s + (1.86 − 3.23i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (−0.654 + 0.377i)7-s + 0.353i·8-s + (0.158 − 0.273i)10-s + (−1.68 − 0.974i)11-s + (0.277 − 0.960i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (1.48 − 0.856i)19-s + (0.193 − 0.111i)20-s + (−0.689 − 1.19i)22-s + (0.389 − 0.673i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.479160334\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479160334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.59 + 3.23i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.46 + 3.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.133 + 0.232i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (-7.96 - 4.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.73 + i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.96 + 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.53iT - 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 + (7.33 - 4.23i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.19 - 9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.92 + 5.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.7 + 6.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 8.92iT - 83T^{2} \) |
| 89 | \( 1 + (-0.464 - 0.267i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.464 + 0.267i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.526061882375803740841809372095, −8.651899310974210326949188784055, −7.915078872514810074591563557058, −7.12422473548658492877037738453, −5.98145080846933360432911491234, −5.38896975615607281424175572036, −4.67950811094152170966627093169, −3.09536516962179707959667223381, −2.78660213021323231316526761286, −0.49960030530360120169901087369,
1.69914196548426895204772195208, 2.84514183161256091429447935564, 3.72130937746582561100770843859, 4.73696865178550778279771079236, 5.63941808434867767800243329490, 6.59009554333405846030602547046, 7.34653466108674566726061097254, 8.129244151766915709580112428078, 9.621966532351997157303673096828, 9.891843289933341365350299688936