L(s) = 1 | + (−0.989 + 0.142i)3-s + (0.841 − 0.540i)5-s + (0.540 + 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.989 + 0.142i)13-s + (−0.755 + 0.654i)15-s + (0.654 − 0.755i)17-s + (−0.281 − 0.959i)19-s + (−0.654 − 0.755i)21-s + (−0.281 − 0.959i)23-s + (0.415 − 0.909i)25-s + (−0.909 + 0.415i)27-s + (−0.540 − 0.841i)29-s + (−0.281 + 0.959i)31-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)3-s + (0.841 − 0.540i)5-s + (0.540 + 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.989 + 0.142i)13-s + (−0.755 + 0.654i)15-s + (0.654 − 0.755i)17-s + (−0.281 − 0.959i)19-s + (−0.654 − 0.755i)21-s + (−0.281 − 0.959i)23-s + (0.415 − 0.909i)25-s + (−0.909 + 0.415i)27-s + (−0.540 − 0.841i)29-s + (−0.281 + 0.959i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.129711439 - 0.9769358257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.129711439 - 0.9769358257i\) |
\(L(1)\) |
\(\approx\) |
\(0.9617673838 - 0.1102319088i\) |
\(L(1)\) |
\(\approx\) |
\(0.9617673838 - 0.1102319088i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.540 + 0.841i)T \) |
| 11 | \( 1 + (0.841 + 0.540i)T \) |
| 13 | \( 1 + (-0.989 + 0.142i)T \) |
| 17 | \( 1 + (0.654 - 0.755i)T \) |
| 19 | \( 1 + (-0.281 - 0.959i)T \) |
| 23 | \( 1 + (-0.281 - 0.959i)T \) |
| 29 | \( 1 + (-0.540 - 0.841i)T \) |
| 31 | \( 1 + (-0.281 + 0.959i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.989 + 0.142i)T \) |
| 43 | \( 1 + (0.540 - 0.841i)T \) |
| 47 | \( 1 + (-0.142 + 0.989i)T \) |
| 53 | \( 1 + (-0.142 - 0.989i)T \) |
| 59 | \( 1 + (-0.989 - 0.142i)T \) |
| 61 | \( 1 + (-0.909 + 0.415i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (-0.959 - 0.281i)T \) |
| 79 | \( 1 + (-0.959 - 0.281i)T \) |
| 83 | \( 1 + (0.755 + 0.654i)T \) |
| 97 | \( 1 + (0.841 - 0.540i)T \) |
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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
−22.56062578089818671032093432875, −21.75485323424757042578704948518, −21.32415038842126830356783190199, −20.16125815239750174525330010613, −19.13733057634644543853179589988, −18.42383512283679412193525995516, −17.41899407407746511203771986193, −17.057692141937810692745138130354, −16.46350679039587919839587470444, −14.96963364932677840740706301085, −14.36108720289885552417380424014, −13.503662843521498053728509793919, −12.57294491639180610995797679184, −11.64942736188999653393364648606, −10.84095081579775586953129232922, −10.176036592394655550219866534329, −9.420211527139878147348841144515, −7.85861944966028665737757059537, −7.18215408420322856587412668940, −6.14275560449184426898834333261, −5.596980984657680226578021475307, −4.44347849835647196636339400576, −3.44858974235722345669367601402, −1.83525085013077083515104938060, −1.13864558130638451201279891918,
0.417276192150597174126118725446, 1.6285061533589318843526884503, 2.52384532326275365708460743237, 4.369950403994835225236489866937, 4.956257462856063263054518187635, 5.75139008320881430430361053363, 6.61936553949247254114423934062, 7.61230033453288697010267659402, 9.09779380344313096995261472460, 9.44837186013171655030933598935, 10.48743790369961940159954811806, 11.50450182937203273189687648590, 12.29540637080879207566941768693, 12.69981902424017632370442734179, 14.08468324240284217313203266601, 14.766737457191431497231452282548, 15.81283469511580490034390878755, 16.653587618157084815005440783504, 17.41557264601859672469972388174, 17.84024631798485529066968186671, 18.75107207580574976127735020967, 19.825375785371841534233594012173, 20.91012860925693687140590951899, 21.45012472530144185431573677918, 22.22014956618461647322420981624