| L(s) = 1 | − 2.09e6·4-s + 1.12e9·7-s − 3.70e11·13-s + 4.39e12·16-s − 3.55e13·19-s − 4.76e14·25-s − 2.35e15·28-s − 9.04e15·31-s − 5.77e16·37-s + 2.65e17·43-s + 7.04e17·49-s + 7.76e17·52-s − 1.08e19·61-s − 9.22e18·64-s + 6.94e18·67-s − 3.90e19·73-s + 7.45e19·76-s + 1.68e20·79-s − 4.15e20·91-s − 1.13e21·97-s + 1.00e21·100-s − 2.44e21·103-s + 4.46e21·109-s + 4.94e21·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.50·7-s − 0.744·13-s + 16-s − 1.32·19-s − 25-s − 1.50·28-s − 1.98·31-s − 1.97·37-s + 1.87·43-s + 1.26·49-s + 0.744·52-s − 1.94·61-s − 64-s + 0.465·67-s − 1.06·73-s + 1.32·76-s + 1.99·79-s − 1.11·91-s − 1.55·97-s + 100-s − 1.79·103-s + 1.80·109-s + 1.50·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{21} T^{2} \) |
| 5 | \( 1 + p^{21} T^{2} \) |
| 7 | \( 1 - 1123983020 T + p^{21} T^{2} \) |
| 11 | \( 1 + p^{21} T^{2} \) |
| 13 | \( 1 + 370076825230 T + p^{21} T^{2} \) |
| 17 | \( 1 + p^{21} T^{2} \) |
| 19 | \( 1 + 35540635313176 T + p^{21} T^{2} \) |
| 23 | \( 1 + p^{21} T^{2} \) |
| 29 | \( 1 + p^{21} T^{2} \) |
| 31 | \( 1 + 9040072142371732 T + p^{21} T^{2} \) |
| 37 | \( 1 + 57776323439003290 T + p^{21} T^{2} \) |
| 41 | \( 1 + p^{21} T^{2} \) |
| 43 | \( 1 - 265258444413820520 T + p^{21} T^{2} \) |
| 47 | \( 1 + p^{21} T^{2} \) |
| 53 | \( 1 + p^{21} T^{2} \) |
| 59 | \( 1 + p^{21} T^{2} \) |
| 61 | \( 1 + 10860691764464843938 T + p^{21} T^{2} \) |
| 67 | \( 1 - 6944332370266921520 T + p^{21} T^{2} \) |
| 71 | \( 1 + p^{21} T^{2} \) |
| 73 | \( 1 + 39098623343480501290 T + p^{21} T^{2} \) |
| 79 | \( 1 - \)\(16\!\cdots\!96\)\( T + p^{21} T^{2} \) |
| 83 | \( 1 + p^{21} T^{2} \) |
| 89 | \( 1 + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(11\!\cdots\!30\)\( T + p^{21} T^{2} \) |
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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
−14.95186301139230229602599090537, −13.99778428686472778004975206547, −12.40369048061808360773761215664, −10.72996068067356596917389199249, −9.000902865703866573097021703985, −7.73336406213411527525924249309, −5.33605102371365360949596689450, −4.14797589156354271684795216600, −1.80245821621307397821780599693, 0,
1.80245821621307397821780599693, 4.14797589156354271684795216600, 5.33605102371365360949596689450, 7.73336406213411527525924249309, 9.000902865703866573097021703985, 10.72996068067356596917389199249, 12.40369048061808360773761215664, 13.99778428686472778004975206547, 14.95186301139230229602599090537
