L(s) = 1 | + (−0.443 − 0.896i)2-s + (−0.0724 − 0.997i)3-s + (−0.607 + 0.794i)4-s + (0.644 + 0.764i)5-s + (−0.861 + 0.506i)6-s + (−0.995 + 0.0965i)7-s + (0.981 + 0.192i)8-s + (−0.989 + 0.144i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.399 − 0.916i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.262 − 0.964i)16-s + (0.779 + 0.626i)17-s + (0.568 + 0.822i)18-s + ⋯ |
L(s) = 1 | + (−0.443 − 0.896i)2-s + (−0.0724 − 0.997i)3-s + (−0.607 + 0.794i)4-s + (0.644 + 0.764i)5-s + (−0.861 + 0.506i)6-s + (−0.995 + 0.0965i)7-s + (0.981 + 0.192i)8-s + (−0.989 + 0.144i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.399 − 0.916i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.262 − 0.964i)16-s + (0.779 + 0.626i)17-s + (0.568 + 0.822i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4225976917 + 0.1543964599i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4225976917 + 0.1543964599i\) |
\(L(1)\) |
\(\approx\) |
\(0.5916643046 - 0.2882123182i\) |
\(L(1)\) |
\(\approx\) |
\(0.5916643046 - 0.2882123182i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.443 - 0.896i)T \) |
| 3 | \( 1 + (-0.0724 - 0.997i)T \) |
| 5 | \( 1 + (0.644 + 0.764i)T \) |
| 7 | \( 1 + (-0.995 + 0.0965i)T \) |
| 13 | \( 1 + (-0.399 - 0.916i)T \) |
| 17 | \( 1 + (0.779 + 0.626i)T \) |
| 19 | \( 1 + (-0.354 - 0.935i)T \) |
| 23 | \( 1 + (0.906 + 0.421i)T \) |
| 29 | \( 1 + (-0.989 - 0.144i)T \) |
| 31 | \( 1 + (0.681 + 0.732i)T \) |
| 37 | \( 1 + (-0.0241 + 0.999i)T \) |
| 41 | \( 1 + (0.262 - 0.964i)T \) |
| 43 | \( 1 + (-0.926 + 0.377i)T \) |
| 47 | \( 1 + (-0.885 - 0.464i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.998 - 0.0483i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + (-0.926 - 0.377i)T \) |
| 71 | \( 1 + (-0.120 - 0.992i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.970 - 0.239i)T \) |
| 83 | \( 1 + (-0.943 + 0.331i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.215 + 0.976i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.6953386404089321263796389471, −19.86828360559351280007453224182, −19.042269275139177844093272508251, −18.33393723469616762820763873819, −17.061372362265392670579972062461, −16.82616321053406491177617079499, −16.329622150610654837469724190667, −15.60146046596894616921754760901, −14.62095913042648100306366975821, −14.107626161143313200815847474857, −13.1927434081697716530555450943, −12.360293797523301883378025990044, −11.19435377027433578397585753960, −10.12775526751194661919316939551, −9.65424068685932606725381341870, −9.1971201312921738990196175078, −8.390081319721554359742021302700, −7.31982811477056672633673814706, −6.250537729962712607328359565437, −5.76032789427358023368856466326, −4.821710221721045799762950809788, −4.19825979258165792558626370358, −3.010642662081060017834153583159, −1.61900652562456362955343491185, −0.218268622573170501924220867681,
1.118091910719240657607690252493, 2.0920365118539142183609813861, 3.05539373786535906314532879880, 3.27870520895146729421669749465, 5.028747269373594007730795882928, 5.94918269171545465628239126079, 6.8413991699886658698914167763, 7.48687700401669756694086313609, 8.44762160564281370378161978728, 9.33517211584181630524976491885, 10.10486636474276161472542479929, 10.76327410382249635631857474834, 11.61420100901477471371361880784, 12.493566343076646106170905501690, 13.11932112789349753814705530935, 13.51702521911698773365665173163, 14.55080335476098508315789503096, 15.421792282873387969758699149145, 16.90144598537208309986561136902, 17.161160748206217792164999624973, 18.04248541078339231107884300031, 18.60495191542838784440760116024, 19.43714176626458406771647851044, 19.58574477189803940744053859862, 20.72966209393275355182494945700