| L(s) = 1 | − 2.84e3·2-s − 5.90e4·3-s + 5.99e6·4-s + 3.10e6·5-s + 1.67e8·6-s + 3.63e8·7-s − 1.10e10·8-s + 3.48e9·9-s − 8.84e9·10-s + 1.45e10·11-s − 3.53e11·12-s + 1.13e11·13-s − 1.03e12·14-s − 1.83e11·15-s + 1.89e13·16-s − 8.58e12·17-s − 9.91e12·18-s − 2.92e13·19-s + 1.86e13·20-s − 2.14e13·21-s − 4.14e13·22-s − 1.55e14·23-s + 6.53e14·24-s − 4.67e14·25-s − 3.22e14·26-s − 2.05e14·27-s + 2.17e15·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.577·3-s + 2.85·4-s + 0.142·5-s + 1.13·6-s + 0.486·7-s − 3.64·8-s + 1/3·9-s − 0.279·10-s + 0.169·11-s − 1.64·12-s + 0.228·13-s − 0.954·14-s − 0.0822·15-s + 4.30·16-s − 1.03·17-s − 0.654·18-s − 1.09·19-s + 0.406·20-s − 0.280·21-s − 0.332·22-s − 0.784·23-s + 2.10·24-s − 0.979·25-s − 0.447·26-s − 0.192·27-s + 1.38·28-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 | \( 1 + 711 p^{2} T + p^{21} T^{2} \) |
| 5 | \( 1 - 124398 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 - 51900560 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 1325621196 p T + p^{21} T^{2} \) |
| 13 | \( 1 - 113350790702 T + p^{21} T^{2} \) |
| 17 | \( 1 + 505258211646 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 1536996803884 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 155899214954280 T + p^{21} T^{2} \) |
| 29 | \( 1 - 2400788707090758 T + p^{21} T^{2} \) |
| 31 | \( 1 - 2239820676947000 T + p^{21} T^{2} \) |
| 37 | \( 1 + 30785069383298890 T + p^{21} T^{2} \) |
| 41 | \( 1 + 103207571041281030 T + p^{21} T^{2} \) |
| 43 | \( 1 + 165557270617488124 T + p^{21} T^{2} \) |
| 47 | \( 1 + 66587216226477408 T + p^{21} T^{2} \) |
| 53 | \( 1 - 435422766592881630 T + p^{21} T^{2} \) |
| 59 | \( 1 - 5534365798259081316 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7176205164722961202 T + p^{21} T^{2} \) |
| 67 | \( 1 + 15755449453068299812 T + p^{21} T^{2} \) |
| 71 | \( 1 - 26457854874259376232 T + p^{21} T^{2} \) |
| 73 | \( 1 - 13471249335464801450 T + p^{21} T^{2} \) |
| 79 | \( 1 + 16886125085525986840 T + p^{21} T^{2} \) |
| 83 | \( 1 + \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 + \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \) |
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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
−19.64882535054477525047031686216, −18.10129991399174111376656103593, −17.12657950832048419183062978110, −15.57887440143012401593974921755, −11.67533953502725227132355006764, −10.27577724529991206035988500531, −8.434446208305040756622708334893, −6.52657195071322903272764533362, −1.80521254452922275171971322257, 0,
1.80521254452922275171971322257, 6.52657195071322903272764533362, 8.434446208305040756622708334893, 10.27577724529991206035988500531, 11.67533953502725227132355006764, 15.57887440143012401593974921755, 17.12657950832048419183062978110, 18.10129991399174111376656103593, 19.64882535054477525047031686216
