| L(s) = 1 | − 32·2-s − 396·3-s + 1.02e3·4-s + 7.35e3·5-s + 1.26e4·6-s + 1.68e4·7-s − 3.27e4·8-s − 2.03e4·9-s − 2.35e5·10-s − 1.08e5·11-s − 4.05e5·12-s − 6.35e5·13-s − 5.37e5·14-s − 2.91e6·15-s + 1.04e6·16-s − 9.22e6·17-s + 6.50e5·18-s − 7.55e6·19-s + 7.52e6·20-s − 6.65e6·21-s + 3.48e6·22-s + 2.64e7·23-s + 1.29e7·24-s + 5.19e6·25-s + 2.03e7·26-s + 7.82e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.940·3-s + 1/2·4-s + 1.05·5-s + 0.665·6-s + 0.377·7-s − 0.353·8-s − 0.114·9-s − 0.743·10-s − 0.203·11-s − 0.470·12-s − 0.474·13-s − 0.267·14-s − 0.989·15-s + 1/4·16-s − 1.57·17-s + 0.0811·18-s − 0.700·19-s + 0.525·20-s − 0.355·21-s + 0.144·22-s + 0.858·23-s + 0.332·24-s + 0.106·25-s + 0.335·26-s + 1.04·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{5} T \) |
| 7 | \( 1 - p^{5} T \) |
| good | 3 | \( 1 + 44 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 - 294 p^{2} T + p^{11} T^{2} \) |
| 11 | \( 1 + 108780 T + p^{11} T^{2} \) |
| 13 | \( 1 + 635842 T + p^{11} T^{2} \) |
| 17 | \( 1 + 9225918 T + p^{11} T^{2} \) |
| 19 | \( 1 + 7555372 T + p^{11} T^{2} \) |
| 23 | \( 1 - 26489400 T + p^{11} T^{2} \) |
| 29 | \( 1 + 169827594 T + p^{11} T^{2} \) |
| 31 | \( 1 + 51362704 T + p^{11} T^{2} \) |
| 37 | \( 1 + 251605906 T + p^{11} T^{2} \) |
| 41 | \( 1 + 928817814 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1818895756 T + p^{11} T^{2} \) |
| 47 | \( 1 - 523343136 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4199520078 T + p^{11} T^{2} \) |
| 59 | \( 1 - 154917444 p T + p^{11} T^{2} \) |
| 61 | \( 1 + 6639312802 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2878139188 T + p^{11} T^{2} \) |
| 71 | \( 1 + 4345596360 T + p^{11} T^{2} \) |
| 73 | \( 1 - 23450332826 T + p^{11} T^{2} \) |
| 79 | \( 1 + 28761853648 T + p^{11} T^{2} \) |
| 83 | \( 1 + 5577757548 T + p^{11} T^{2} \) |
| 89 | \( 1 - 78002173386 T + p^{11} T^{2} \) |
| 97 | \( 1 + 26685859630 T + p^{11} T^{2} \) |
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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
−16.81738465820805203026150565486, −15.04233000935699363687852661696, −13.24588520763136530387328541602, −11.51507282582037421885839979568, −10.41056014376734769663283150947, −8.860882965618912621232306351526, −6.71940247350786512013959615006, −5.29804417707519144707909944559, −2.01822066300952997483034591862, 0,
2.01822066300952997483034591862, 5.29804417707519144707909944559, 6.71940247350786512013959615006, 8.860882965618912621232306351526, 10.41056014376734769663283150947, 11.51507282582037421885839979568, 13.24588520763136530387328541602, 15.04233000935699363687852661696, 16.81738465820805203026150565486
