| L(s) = 1 | − 2.86e3·5-s − 9.12e3·7-s + 6.68e5·11-s + 2.05e6·13-s − 1.60e6·17-s + 2.30e5·19-s − 4.30e7·23-s − 4.06e7·25-s + 1.41e8·29-s − 2.33e8·31-s + 2.61e7·35-s + 2.78e8·37-s + 1.18e9·41-s − 8.56e8·43-s − 1.66e9·47-s − 1.89e9·49-s + 3.85e9·53-s − 1.91e9·55-s + 1.03e10·59-s + 1.85e8·61-s − 5.87e9·65-s − 2.91e9·67-s + 1.26e10·71-s − 1.52e10·73-s − 6.09e9·77-s + 3.66e10·79-s − 9.21e9·83-s + ⋯ |
| L(s) = 1 | − 0.409·5-s − 0.205·7-s + 1.25·11-s + 1.53·13-s − 0.274·17-s + 0.0213·19-s − 1.39·23-s − 0.832·25-s + 1.28·29-s − 1.46·31-s + 0.0840·35-s + 0.659·37-s + 1.59·41-s − 0.888·43-s − 1.05·47-s − 0.957·49-s + 1.26·53-s − 0.512·55-s + 1.88·59-s + 0.0281·61-s − 0.628·65-s − 0.263·67-s + 0.832·71-s − 0.858·73-s − 0.256·77-s + 1.33·79-s − 0.256·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.135953201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.135953201\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2862 T + p^{11} T^{2} \) |
| 7 | \( 1 + 1304 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 668196 T + p^{11} T^{2} \) |
| 13 | \( 1 - 2052950 T + p^{11} T^{2} \) |
| 17 | \( 1 + 1604178 T + p^{11} T^{2} \) |
| 19 | \( 1 - 230500 T + p^{11} T^{2} \) |
| 23 | \( 1 + 43012728 T + p^{11} T^{2} \) |
| 29 | \( 1 - 141745194 T + p^{11} T^{2} \) |
| 31 | \( 1 + 233221904 T + p^{11} T^{2} \) |
| 37 | \( 1 - 278269694 T + p^{11} T^{2} \) |
| 41 | \( 1 - 1181577510 T + p^{11} T^{2} \) |
| 43 | \( 1 + 856975172 T + p^{11} T^{2} \) |
| 47 | \( 1 + 35405424 p T + p^{11} T^{2} \) |
| 53 | \( 1 - 3851181666 T + p^{11} T^{2} \) |
| 59 | \( 1 - 10339000596 T + p^{11} T^{2} \) |
| 61 | \( 1 - 185948102 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2915010572 T + p^{11} T^{2} \) |
| 71 | \( 1 - 12662314200 T + p^{11} T^{2} \) |
| 73 | \( 1 + 15201270694 T + p^{11} T^{2} \) |
| 79 | \( 1 - 36644027488 T + p^{11} T^{2} \) |
| 83 | \( 1 + 9217637028 T + p^{11} T^{2} \) |
| 89 | \( 1 + 30573828810 T + p^{11} T^{2} \) |
| 97 | \( 1 - 145701815906 T + p^{11} T^{2} \) |
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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
−11.20120955043241429575775090604, −9.953568169637060501820027009927, −8.898251692004170065272274042951, −8.011196407829107622703419346921, −6.67040177993114394753782060489, −5.87591849691252862985346379658, −4.20140958776355222916800889442, −3.53167042719154744058912538288, −1.86685531469641935671079688133, −0.70982672012471692932049268666,
0.70982672012471692932049268666, 1.86685531469641935671079688133, 3.53167042719154744058912538288, 4.20140958776355222916800889442, 5.87591849691252862985346379658, 6.67040177993114394753782060489, 8.011196407829107622703419346921, 8.898251692004170065272274042951, 9.953568169637060501820027009927, 11.20120955043241429575775090604
