L(s) = 1 | + (−0.281 − 0.959i)3-s + (0.415 − 0.909i)5-s + (−0.909 − 0.415i)7-s + (−0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (0.281 + 0.959i)13-s + (−0.989 − 0.142i)15-s + (0.142 + 0.989i)17-s + (−0.540 − 0.841i)19-s + (−0.142 + 0.989i)21-s + (−0.540 − 0.841i)23-s + (−0.654 − 0.755i)25-s + (0.755 + 0.654i)27-s + (−0.909 − 0.415i)29-s + (−0.540 + 0.841i)31-s + ⋯ |
L(s) = 1 | + (−0.281 − 0.959i)3-s + (0.415 − 0.909i)5-s + (−0.909 − 0.415i)7-s + (−0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (0.281 + 0.959i)13-s + (−0.989 − 0.142i)15-s + (0.142 + 0.989i)17-s + (−0.540 − 0.841i)19-s + (−0.142 + 0.989i)21-s + (−0.540 − 0.841i)23-s + (−0.654 − 0.755i)25-s + (0.755 + 0.654i)27-s + (−0.909 − 0.415i)29-s + (−0.540 + 0.841i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1461890771 - 0.2734318221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1461890771 - 0.2734318221i\) |
\(L(1)\) |
\(\approx\) |
\(0.5913351653 - 0.3829353628i\) |
\(L(1)\) |
\(\approx\) |
\(0.5913351653 - 0.3829353628i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.909 - 0.415i)T \) |
| 11 | \( 1 + (-0.415 - 0.909i)T \) |
| 13 | \( 1 + (0.281 + 0.959i)T \) |
| 17 | \( 1 + (0.142 + 0.989i)T \) |
| 19 | \( 1 + (-0.540 - 0.841i)T \) |
| 23 | \( 1 + (-0.540 - 0.841i)T \) |
| 29 | \( 1 + (-0.909 - 0.415i)T \) |
| 31 | \( 1 + (-0.540 + 0.841i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.281 + 0.959i)T \) |
| 43 | \( 1 + (-0.909 + 0.415i)T \) |
| 47 | \( 1 + (0.959 + 0.281i)T \) |
| 53 | \( 1 + (-0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.281 + 0.959i)T \) |
| 61 | \( 1 + (-0.755 - 0.654i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.841 + 0.540i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (0.989 - 0.142i)T \) |
| 97 | \( 1 + (0.415 - 0.909i)T \) |
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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
−22.85118538837622539412412134526, −22.329155082958344564040038167053, −21.75771485457339133986334703854, −20.62171025276427512974995409560, −20.21670514684324454864153250544, −18.86874145532042595571808087199, −18.27354595417318358702943557799, −17.40031736290279283746856649148, −16.55294481653045203682698274294, −15.44195706739479096867139314279, −15.294626648548028951350198814976, −14.230854238612571536210954324492, −13.23243447385897060114258437599, −12.27800932450580437830500444437, −11.32677595420090949425513225204, −10.33524304331530218038597627146, −9.9082620224559477589790360247, −9.174259252740273862229331649868, −7.83621113322698264628357826546, −6.79903083927598789286510770408, −5.81419246131036894159080265522, −5.26283900879891489061952690040, −3.80518318097986999740054297281, −3.13606338503143907214714832583, −2.12677467810815140398424580257,
0.1487552814623217238071425555, 1.36692337681245281787008240144, 2.380935969673286404929942655748, 3.68147745222083930371739787660, 4.84557765698301519296174752161, 6.06755108676641170098447721722, 6.36228922653044967807393560294, 7.58426602361037614717664287134, 8.56080742567143864392497613196, 9.192624226041596603055066437372, 10.44791990218148795498390985572, 11.251283689161453532188519374770, 12.38360321750401885771231694196, 12.94986271362709655537525500876, 13.55594262568465173654659265819, 14.30701487397596768458503638449, 15.78950419075264191481726310680, 16.70537759161675832716828663041, 16.88358963579257299971769892380, 18.04585796433458336938935762111, 18.87722596222796523943209032571, 19.55874072744277685436478542013, 20.23708743929627071620669786467, 21.4095561664601078462515525929, 21.95391658589791548626353727230