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 | \( 1 + (-0.215 - 0.976i)T \) |
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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
−20.72966209393275355182494945700, −19.58574477189803940744053859862, −19.43714176626458406771647851044, −18.60495191542838784440760116024, −18.04248541078339231107884300031, −17.161160748206217792164999624973, −16.90144598537208309986561136902, −15.421792282873387969758699149145, −14.55080335476098508315789503096, −13.51702521911698773365665173163, −13.11932112789349753814705530935, −12.493566343076646106170905501690, −11.61420100901477471371361880784, −10.76327410382249635631857474834, −10.10486636474276161472542479929, −9.33517211584181630524976491885, −8.44762160564281370378161978728, −7.48687700401669756694086313609, −6.8413991699886658698914167763, −5.94918269171545465628239126079, −5.028747269373594007730795882928, −3.27870520895146729421669749465, −3.05539373786535906314532879880, −2.0920365118539142183609813861, −1.118091910719240657607690252493,
0.218268622573170501924220867681, 1.61900652562456362955343491185, 3.010642662081060017834153583159, 4.19825979258165792558626370358, 4.821710221721045799762950809788, 5.76032789427358023368856466326, 6.250537729962712607328359565437, 7.31982811477056672633673814706, 8.390081319721554359742021302700, 9.1971201312921738990196175078, 9.65424068685932606725381341870, 10.12775526751194661919316939551, 11.19435377027433578397585753960, 12.360293797523301883378025990044, 13.1927434081697716530555450943, 14.107626161143313200815847474857, 14.62095913042648100306366975821, 15.60146046596894616921754760901, 16.329622150610654837469724190667, 16.82616321053406491177617079499, 17.061372362265392670579972062461, 18.33393723469616762820763873819, 19.042269275139177844093272508251, 19.86828360559351280007453224182, 20.6953386404089321263796389471