| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 3.63e3·5-s + 3.29e4·7-s − 3.27e4·8-s + 1.16e5·10-s + 7.58e5·11-s − 2.48e6·13-s − 1.05e6·14-s + 1.04e6·16-s − 8.29e6·17-s − 1.08e7·19-s − 3.71e6·20-s − 2.42e7·22-s − 2.05e7·23-s − 3.56e7·25-s + 7.94e7·26-s + 3.37e7·28-s − 2.88e7·29-s + 1.50e8·31-s − 3.35e7·32-s + 2.65e8·34-s − 1.19e8·35-s − 3.19e8·37-s + 3.47e8·38-s + 1.18e8·40-s + 3.68e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.519·5-s + 0.740·7-s − 0.353·8-s + 0.367·10-s + 1.42·11-s − 1.85·13-s − 0.523·14-s + 1/4·16-s − 1.41·17-s − 1.00·19-s − 0.259·20-s − 1.00·22-s − 0.665·23-s − 0.730·25-s + 1.31·26-s + 0.370·28-s − 0.260·29-s + 0.944·31-s − 0.176·32-s + 1.00·34-s − 0.384·35-s − 0.758·37-s + 0.711·38-s + 0.183·40-s + 0.496·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 726 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 32936 T + p^{11} T^{2} \) |
| 11 | \( 1 - 758748 T + p^{11} T^{2} \) |
| 13 | \( 1 + 2482858 T + p^{11} T^{2} \) |
| 17 | \( 1 + 8290386 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10867300 T + p^{11} T^{2} \) |
| 23 | \( 1 + 20539272 T + p^{11} T^{2} \) |
| 29 | \( 1 + 28814550 T + p^{11} T^{2} \) |
| 31 | \( 1 - 150501392 T + p^{11} T^{2} \) |
| 37 | \( 1 + 8645722 p T + p^{11} T^{2} \) |
| 41 | \( 1 - 368008998 T + p^{11} T^{2} \) |
| 43 | \( 1 - 620469572 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2763110256 T + p^{11} T^{2} \) |
| 53 | \( 1 - 268284258 T + p^{11} T^{2} \) |
| 59 | \( 1 + 1672894740 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7787197498 T + p^{11} T^{2} \) |
| 67 | \( 1 - 18706694156 T + p^{11} T^{2} \) |
| 71 | \( 1 - 8346990888 T + p^{11} T^{2} \) |
| 73 | \( 1 - 19641746522 T + p^{11} T^{2} \) |
| 79 | \( 1 + 5873807200 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8492558172 T + p^{11} T^{2} \) |
| 89 | \( 1 + 75527864010 T + p^{11} T^{2} \) |
| 97 | \( 1 + 82356782494 T + p^{11} T^{2} \) |
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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
−15.38676241250759586377374100276, −14.35885358090150548084475966669, −12.22883928367151759071501010853, −11.22510283957828840396429645474, −9.568452712846735819717320593072, −8.168563607164780177940211680212, −6.74458678728213738616867926991, −4.40165481434889541833890815260, −2.00592727109788167431861940869, 0,
2.00592727109788167431861940869, 4.40165481434889541833890815260, 6.74458678728213738616867926991, 8.168563607164780177940211680212, 9.568452712846735819717320593072, 11.22510283957828840396429645474, 12.22883928367151759071501010853, 14.35885358090150548084475966669, 15.38676241250759586377374100276
