L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.104 − 0.994i)11-s + (0.669 + 0.743i)12-s + (−0.978 + 0.207i)13-s + (−0.5 − 0.866i)14-s + (0.913 − 0.406i)16-s + (0.309 − 0.951i)17-s + (0.913 + 0.406i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.104 − 0.994i)11-s + (0.669 + 0.743i)12-s + (−0.978 + 0.207i)13-s + (−0.5 − 0.866i)14-s + (0.913 − 0.406i)16-s + (0.309 − 0.951i)17-s + (0.913 + 0.406i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05323506464 + 0.004296934922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05323506464 + 0.004296934922i\) |
\(L(1)\) |
\(\approx\) |
\(0.4243234904 - 0.5045716014i\) |
\(L(1)\) |
\(\approx\) |
\(0.4243234904 - 0.5045716014i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.913 - 0.406i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.978 + 0.207i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.669 + 0.743i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.104 - 0.994i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.38499573869883599573971741359, −17.06550228210996582781696321262, −16.574874446134152318178480770931, −15.382873675026482409640202699917, −15.188604264997459252239770701375, −14.857142094337575394788335123732, −14.08278599054786267226936069740, −13.18810156511398357130270536984, −12.16491841584634530384998945888, −12.05852714370197387384386985177, −10.83825934187312186714611269860, −10.21226449375038032387808635608, −9.78039263525124080069298607372, −8.93602012637431940930190459399, −8.33295787063253087929620417974, −7.616005632596089692702977337198, −6.95082017441406945233693978387, −5.91080231994397580812248203641, −5.60726212979462364007527981158, −4.76117948257670465861640882648, −4.3308073338451793695041235782, −3.63874895063580661413433004907, −2.32440780679237984416986568112, −1.4681967203526161749812962737, −0.01952821129266723059230610464,
0.77170537549155562431192279707, 1.61171453476107334390668641820, 2.32931105204126709056967220403, 2.94030513067582006623459948543, 4.06022138144928763120582411587, 4.77656572865928795510176341361, 5.318470142585708721988953296297, 6.1679129146872586981080561809, 7.09849900601245417735713543500, 7.843220612608013847111131497863, 8.27897046211127741851791226332, 9.04986387675750760302205646785, 9.995121969534940570608901404815, 10.63210688722528666926088746901, 11.36222076143747443802190543918, 11.58138708610619040146045454156, 12.408178532523665803974007998504, 12.96936246920455776210448664984, 13.6851847835839045097297682161, 14.30612324831328672551937440573, 14.6101589176144926525752564545, 16.0533679191932954628688648215, 16.73286646949290745726104818651, 17.278619397477720452573065735, 17.8009386925391545521992293510