| L(s) = 1 | + 34·2-s − 892·4-s + 1.75e4·7-s − 9.99e4·8-s + 4.68e5·11-s + 3.74e5·13-s + 5.96e5·14-s − 1.57e6·16-s − 3.72e6·17-s − 3.79e5·19-s + 1.59e7·22-s − 3.24e7·23-s + 1.27e7·26-s − 1.56e7·28-s − 6.96e7·29-s + 1.71e8·31-s + 1.51e8·32-s − 1.26e8·34-s + 2.91e8·37-s − 1.29e7·38-s − 1.91e8·41-s + 1.75e9·43-s − 4.18e8·44-s − 1.10e9·46-s + 1.62e9·47-s − 1.66e9·49-s − 3.33e8·52-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 0.435·4-s + 0.394·7-s − 1.07·8-s + 0.877·11-s + 0.279·13-s + 0.296·14-s − 0.374·16-s − 0.636·17-s − 0.0351·19-s + 0.659·22-s − 1.05·23-s + 0.209·26-s − 0.171·28-s − 0.630·29-s + 1.07·31-s + 0.796·32-s − 0.477·34-s + 0.690·37-s − 0.0264·38-s − 0.257·41-s + 1.82·43-s − 0.382·44-s − 0.790·46-s + 1.03·47-s − 0.844·49-s − 0.121·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 17 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 2508 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 468788 T + p^{11} T^{2} \) |
| 13 | \( 1 - 374042 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3724286 T + p^{11} T^{2} \) |
| 19 | \( 1 + 379460 T + p^{11} T^{2} \) |
| 23 | \( 1 + 32458092 T + p^{11} T^{2} \) |
| 29 | \( 1 + 69696710 T + p^{11} T^{2} \) |
| 31 | \( 1 - 171448632 T + p^{11} T^{2} \) |
| 37 | \( 1 - 291340546 T + p^{11} T^{2} \) |
| 41 | \( 1 + 191343242 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1759857392 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1623469924 T + p^{11} T^{2} \) |
| 53 | \( 1 + 644888642 T + p^{11} T^{2} \) |
| 59 | \( 1 + 925569220 T + p^{11} T^{2} \) |
| 61 | \( 1 + 10898589338 T + p^{11} T^{2} \) |
| 67 | \( 1 + 3795674064 T + p^{11} T^{2} \) |
| 71 | \( 1 - 22966943728 T + p^{11} T^{2} \) |
| 73 | \( 1 + 9880820458 T + p^{11} T^{2} \) |
| 79 | \( 1 + 20768886240 T + p^{11} T^{2} \) |
| 83 | \( 1 - 3204862008 T + p^{11} T^{2} \) |
| 89 | \( 1 + 63176321130 T + p^{11} T^{2} \) |
| 97 | \( 1 + 126494473874 T + p^{11} T^{2} \) |
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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
−9.654545631614378729517185210299, −8.886922918709171364624397535600, −7.892823322800810683718924780841, −6.49753025694724846106933919702, −5.70255636711372028562311505827, −4.48202430106997520977059397360, −3.91327848022365141866084606521, −2.61470968164577575293272677894, −1.26127240094190220718269134434, 0,
1.26127240094190220718269134434, 2.61470968164577575293272677894, 3.91327848022365141866084606521, 4.48202430106997520977059397360, 5.70255636711372028562311505827, 6.49753025694724846106933919702, 7.892823322800810683718924780841, 8.886922918709171364624397535600, 9.654545631614378729517185210299
