L(s) = 1 | + (0.743 − 0.669i)3-s + (0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (−0.587 + 0.809i)13-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (0.406 − 0.913i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.978 + 0.207i)31-s + (0.743 + 0.669i)33-s + (0.994 + 0.104i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (0.743 − 0.669i)3-s + (0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (−0.587 + 0.809i)13-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (0.406 − 0.913i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.978 + 0.207i)31-s + (0.743 + 0.669i)33-s + (0.994 + 0.104i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.845980358 + 0.3311089107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.845980358 + 0.3311089107i\) |
\(L(1)\) |
\(\approx\) |
\(1.318373241 - 0.05724957234i\) |
\(L(1)\) |
\(\approx\) |
\(1.318373241 - 0.05724957234i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.743 - 0.669i)T \) |
| 11 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.587 + 0.809i)T \) |
| 17 | \( 1 + (-0.207 + 0.978i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (0.406 - 0.913i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.994 + 0.104i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.743 + 0.669i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.207 + 0.978i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.994 - 0.104i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.913 + 0.406i)T \) |
| 97 | \( 1 + (-0.951 - 0.309i)T \) |
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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
−20.72850942643624867375002073731, −19.980700037956990354862003134070, −19.41937616125896129552275950234, −18.6895640820909824318842972737, −17.65298877893533010529602393304, −16.90150923644392764583239910480, −16.03572310301113299732935983918, −15.45626088382600901341787899966, −14.74255855040357117924599712821, −13.85701343856617492990414926047, −13.381392833253021921375237651807, −12.39647036736433044340166440224, −11.252876837475377290882370546877, −10.7522022752302828334435201129, −9.76059179771478481120583253275, −9.12558164659030714231384537479, −8.350782982006693028333226079482, −7.59187445570023470657908512341, −6.6426313771447941355589376261, −5.404219660389931095105695864739, −4.84456779881571510358441275784, −3.741031623623208166896200024, −2.97412307577536003105380405094, −2.25781436207479255660668497952, −0.69738237939354831953127794402,
1.18695454075976092222105127175, 2.12126459995500655009702036039, 2.76677338415168409677640342723, 4.127181758932884904472804318564, 4.55763058789343652959044059295, 6.22567919360754815001152259824, 6.55830898896739636815357150769, 7.68217348828951558547632189470, 8.16681471737729145224797297418, 9.1997687872992365122672379740, 9.78759444772621291154114482533, 10.75549957945445422100946938059, 11.91784267939161452119589694486, 12.4840025669903470978899949868, 13.104075793564917804886993264846, 14.07899140973882689759175074159, 14.7657471650969654798916015629, 15.16957067516543553026446444343, 16.3632104880010978758110389113, 17.24976434063087506065211106115, 17.80450937822709578956458660561, 18.80329910085910413156396479684, 19.30786345404308819170500535731, 19.93629468154976395222465895141, 20.86244091728397295073727354947