L(s) = 1 | + 1.63·2-s − 3-s + 0.674·4-s + (1.84 + 1.25i)5-s − 1.63·6-s + (−1.79 − 1.94i)7-s − 2.16·8-s + 9-s + (3.02 + 2.05i)10-s + (−3.27 + 0.520i)11-s − 0.674·12-s + 4.17i·13-s + (−2.93 − 3.18i)14-s + (−1.84 − 1.25i)15-s − 4.89·16-s + 7.58i·17-s + ⋯ |
L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.337·4-s + (0.826 + 0.562i)5-s − 0.667·6-s + (−0.677 − 0.735i)7-s − 0.766·8-s + 0.333·9-s + (0.956 + 0.650i)10-s + (−0.987 + 0.156i)11-s − 0.194·12-s + 1.15i·13-s + (−0.783 − 0.850i)14-s + (−0.477 − 0.324i)15-s − 1.22·16-s + 1.83i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585206924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585206924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (-1.84 - 1.25i)T \) |
| 7 | \( 1 + (1.79 + 1.94i)T \) |
| 11 | \( 1 + (3.27 - 0.520i)T \) |
good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 13 | \( 1 - 4.17iT - 13T^{2} \) |
| 17 | \( 1 - 7.58iT - 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 2.17iT - 23T^{2} \) |
| 29 | \( 1 - 5.39iT - 29T^{2} \) |
| 31 | \( 1 - 0.864iT - 31T^{2} \) |
| 37 | \( 1 + 2.91iT - 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 5.31T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - 2.25iT - 73T^{2} \) |
| 79 | \( 1 - 8.24iT - 79T^{2} \) |
| 83 | \( 1 - 5.54iT - 83T^{2} \) |
| 89 | \( 1 + 1.62iT - 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−10.00491397104824649547119514675, −9.637167153466399458519963757033, −8.358685911057881779659965165893, −6.96110723173412584779117269149, −6.62597838715978800315433041448, −5.63616780187344535615897547198, −5.07446960138911927029723939221, −3.89099786786973448887172822618, −3.20081032313592026094388905845, −1.78301313899073997006544919031,
0.48308690064655282983783075574, 2.58560545077386014653762266434, 3.17560142585812479185398680543, 4.79481358004456032191477338416, 5.26452413264788982022650780918, 5.77167660399299219085556687708, 6.60562636128856621065011166525, 7.80409889216704082671168349244, 8.903912457042417587459566639704, 9.694360426944226311473522271312