| L(s) = 1 | + 1.72e3·2-s − 5.90e4·3-s + 8.88e5·4-s − 4.15e7·5-s − 1.02e8·6-s + 5.38e8·7-s − 2.08e9·8-s + 3.48e9·9-s − 7.17e10·10-s − 6.41e10·11-s − 5.24e10·12-s − 1.30e11·13-s + 9.30e11·14-s + 2.45e12·15-s − 5.47e12·16-s + 8.24e12·17-s + 6.02e12·18-s + 1.34e13·19-s − 3.68e13·20-s − 3.17e13·21-s − 1.10e14·22-s − 2.33e14·23-s + 1.23e14·24-s + 1.24e15·25-s − 2.26e14·26-s − 2.05e14·27-s + 4.78e14·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.423·4-s − 1.90·5-s − 0.688·6-s + 0.720·7-s − 0.687·8-s + 1/3·9-s − 2.26·10-s − 0.745·11-s − 0.244·12-s − 0.263·13-s + 0.859·14-s + 1.09·15-s − 1.24·16-s + 0.991·17-s + 0.397·18-s + 0.504·19-s − 0.805·20-s − 0.415·21-s − 0.889·22-s − 1.17·23-s + 0.396·24-s + 2.61·25-s − 0.314·26-s − 0.192·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{10} T \) |
| good | 2 | \( 1 - 27 p^{6} T + p^{21} T^{2} \) |
| 5 | \( 1 + 8302554 p T + p^{21} T^{2} \) |
| 7 | \( 1 - 76918544 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 64113040188 T + p^{21} T^{2} \) |
| 13 | \( 1 + 10075392922 p T + p^{21} T^{2} \) |
| 17 | \( 1 - 8242029723618 T + p^{21} T^{2} \) |
| 19 | \( 1 - 710110618580 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 233184825844776 T + p^{21} T^{2} \) |
| 29 | \( 1 + 2024562031123770 T + p^{21} T^{2} \) |
| 31 | \( 1 + 6869194988701768 T + p^{21} T^{2} \) |
| 37 | \( 1 - 3443998107027638 T + p^{21} T^{2} \) |
| 41 | \( 1 + 21842403084625158 T + p^{21} T^{2} \) |
| 43 | \( 1 + 71792816814133756 T + p^{21} T^{2} \) |
| 47 | \( 1 - 283544719418655648 T + p^{21} T^{2} \) |
| 53 | \( 1 + 2172285419049898146 T + p^{21} T^{2} \) |
| 59 | \( 1 - 1534831476719068260 T + p^{21} T^{2} \) |
| 61 | \( 1 - 4311589520797626062 T + p^{21} T^{2} \) |
| 67 | \( 1 - 9243910904037307868 T + p^{21} T^{2} \) |
| 71 | \( 1 + 20387361256404760728 T + p^{21} T^{2} \) |
| 73 | \( 1 - 16617754439328636074 T + p^{21} T^{2} \) |
| 79 | \( 1 - 67940304745507627880 T + p^{21} T^{2} \) |
| 83 | \( 1 - 39503732340682314684 T + p^{21} T^{2} \) |
| 89 | \( 1 - 41611676186839694490 T + p^{21} T^{2} \) |
| 97 | \( 1 - 57181473208903260098 T + p^{21} T^{2} \) |
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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
−20.49998418375541284487510183570, −18.49609057070930816211523714684, −16.01860593020274242393032758594, −14.70527596574691470092859353756, −12.45381657294462324045831931750, −11.37475892508532153114703159453, −7.72929137964893599637201923424, −5.11898319349057759278703990278, −3.69966807572177230845693757257, 0,
3.69966807572177230845693757257, 5.11898319349057759278703990278, 7.72929137964893599637201923424, 11.37475892508532153114703159453, 12.45381657294462324045831931750, 14.70527596574691470092859353756, 16.01860593020274242393032758594, 18.49609057070930816211523714684, 20.49998418375541284487510183570
