| L(s) = 1 | + 1.02e3·2-s − 5.90e4·3-s + 1.04e6·4-s + 1.29e7·5-s − 6.04e7·6-s − 4.79e8·7-s + 1.07e9·8-s + 3.48e9·9-s + 1.32e10·10-s + 1.15e11·11-s − 6.19e10·12-s + 2.95e11·13-s − 4.91e11·14-s − 7.64e11·15-s + 1.09e12·16-s + 6.62e12·17-s + 3.57e12·18-s + 2.85e13·19-s + 1.35e13·20-s + 2.83e13·21-s + 1.18e14·22-s + 3.35e14·23-s − 6.34e13·24-s − 3.09e14·25-s + 3.02e14·26-s − 2.05e14·27-s − 5.02e14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.593·5-s − 0.408·6-s − 0.641·7-s + 0.353·8-s + 1/3·9-s + 0.419·10-s + 1.34·11-s − 0.288·12-s + 0.594·13-s − 0.453·14-s − 0.342·15-s + 1/4·16-s + 0.797·17-s + 0.235·18-s + 1.06·19-s + 0.296·20-s + 0.370·21-s + 0.950·22-s + 1.68·23-s − 0.204·24-s − 0.648·25-s + 0.420·26-s − 0.192·27-s − 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.791246189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.791246189\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{10} T \) |
| 3 | \( 1 + p^{10} T \) |
| good | 5 | \( 1 - 12954174 T + p^{21} T^{2} \) |
| 7 | \( 1 + 68501872 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 115657781700 T + p^{21} T^{2} \) |
| 13 | \( 1 - 22742942054 p T + p^{21} T^{2} \) |
| 17 | \( 1 - 389822554818 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 1504009692884 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 335385196791000 T + p^{21} T^{2} \) |
| 29 | \( 1 + 699224214482106 T + p^{21} T^{2} \) |
| 31 | \( 1 + 3484957262657992 T + p^{21} T^{2} \) |
| 37 | \( 1 + 35181531093012298 T + p^{21} T^{2} \) |
| 41 | \( 1 - 6132056240639994 T + p^{21} T^{2} \) |
| 43 | \( 1 - 233260850096910596 T + p^{21} T^{2} \) |
| 47 | \( 1 + 580205712121346400 T + p^{21} T^{2} \) |
| 53 | \( 1 + 1394471665941750306 T + p^{21} T^{2} \) |
| 59 | \( 1 - 2352476807159705700 T + p^{21} T^{2} \) |
| 61 | \( 1 - 9920628300330384590 T + p^{21} T^{2} \) |
| 67 | \( 1 - 26068981808996843996 T + p^{21} T^{2} \) |
| 71 | \( 1 + 13336955952504341400 T + p^{21} T^{2} \) |
| 73 | \( 1 - 9037529597968684202 T + p^{21} T^{2} \) |
| 79 | \( 1 + 77283864571811027992 T + p^{21} T^{2} \) |
| 83 | \( 1 + \)\(15\!\cdots\!88\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(25\!\cdots\!18\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(10\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.34529784155541816154381778890, −16.09175225932649804357906744342, −14.29509054318518246790951719694, −12.85425731468585343577737145947, −11.38448177123382940722399841726, −9.559390863496340615618524455315, −6.80158946705549857698693651571, −5.50534943306919418336386885848, −3.51945344495609451803952401379, −1.27587786172355853637548451035,
1.27587786172355853637548451035, 3.51945344495609451803952401379, 5.50534943306919418336386885848, 6.80158946705549857698693651571, 9.559390863496340615618524455315, 11.38448177123382940722399841726, 12.85425731468585343577737145947, 14.29509054318518246790951719694, 16.09175225932649804357906744342, 17.34529784155541816154381778890
