L(s) = 1 | + (−0.281 + 0.959i)2-s + (−0.755 − 0.654i)3-s + (−0.841 − 0.540i)4-s + (0.841 − 0.540i)6-s + (−0.909 + 0.415i)7-s + (0.755 − 0.654i)8-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)11-s + (0.281 + 0.959i)12-s + (0.909 + 0.415i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + (−0.989 − 0.142i)18-s + (0.841 + 0.540i)19-s + ⋯ |
L(s) = 1 | + (−0.281 + 0.959i)2-s + (−0.755 − 0.654i)3-s + (−0.841 − 0.540i)4-s + (0.841 − 0.540i)6-s + (−0.909 + 0.415i)7-s + (0.755 − 0.654i)8-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)11-s + (0.281 + 0.959i)12-s + (0.909 + 0.415i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + (−0.989 − 0.142i)18-s + (0.841 + 0.540i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5323706285 + 0.3436096007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5323706285 + 0.3436096007i\) |
\(L(1)\) |
\(\approx\) |
\(0.6460083460 + 0.2392673332i\) |
\(L(1)\) |
\(\approx\) |
\(0.6460083460 + 0.2392673332i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.281 + 0.959i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (-0.909 + 0.415i)T \) |
| 11 | \( 1 + (0.959 - 0.281i)T \) |
| 13 | \( 1 + (0.909 + 0.415i)T \) |
| 17 | \( 1 + (0.540 + 0.841i)T \) |
| 19 | \( 1 + (0.841 + 0.540i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (-0.654 - 0.755i)T \) |
| 37 | \( 1 + (0.989 - 0.142i)T \) |
| 41 | \( 1 + (-0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.909 - 0.415i)T \) |
| 59 | \( 1 + (-0.415 + 0.909i)T \) |
| 61 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.281 - 0.959i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.540 + 0.841i)T \) |
| 79 | \( 1 + (0.415 - 0.909i)T \) |
| 83 | \( 1 + (-0.989 + 0.142i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.989 - 0.142i)T \) |
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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
−29.03714613672458195147746489509, −28.12527214542707831683285217463, −27.35590792736875008289951768528, −26.41117266668051941007129637603, −25.381003124734005638768572908925, −23.48198493989084513952784524759, −22.60338340383062096836505982752, −22.096314199442213125168602188789, −20.731547948425013984772230603037, −20.06431310915466022545223815345, −18.75925331892166261152762451861, −17.69691022515107621219031991543, −16.73212552571081502279641439808, −15.80356193976797582307189205852, −14.116861999503751994528659976872, −12.88016890779948869382274408843, −11.815822286867463879981571330742, −10.89300828157441475517475958708, −9.78161757937078461497544708973, −9.10152475091617229291281522063, −7.16866458457861117229514760283, −5.63292532228803086748475309112, −4.145895252652475294042820317453, −3.22553722307239668937040501599, −0.925643852985296184549135749172,
1.314745966289944636978312823948, 3.86347645729603121328694079926, 5.70479529666232753333742118785, 6.24876654894646372758984495474, 7.399472008128270643695449594384, 8.72783549314365145569507502089, 9.900305759298969661731974344077, 11.364192504779434746015255957797, 12.67892757041488537720291479973, 13.61948411670766248906934016285, 14.87472609557958061305293091019, 16.41988403470798877711864642838, 16.60771872105992809789224826910, 18.08732090809913054894033018548, 18.78492558959707886387089314735, 19.69057237064235244626014246386, 21.79584437076539868975597645470, 22.61352416948139776534428813035, 23.448683121990604189983787670306, 24.45492232806362733831261537631, 25.294098322798898940009332036132, 26.183340307600938198433923722337, 27.59516001453058181444862560123, 28.30108332010842005829290045949, 29.24244376976266581174576970971