| L(s) = 1 | + 4-s + 5·7-s − 2·9-s + 6·11-s − 4·13-s − 3·16-s + 4·17-s + 2·19-s + 3·23-s − 2·25-s + 5·28-s − 5·29-s + 13·31-s − 2·36-s − 9·37-s − 3·41-s + 6·43-s + 6·44-s − 2·47-s + 7·49-s − 4·52-s − 7·53-s + 59-s − 4·61-s − 10·63-s − 7·64-s + 8·67-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.88·7-s − 2/3·9-s + 1.80·11-s − 1.10·13-s − 3/4·16-s + 0.970·17-s + 0.458·19-s + 0.625·23-s − 2/5·25-s + 0.944·28-s − 0.928·29-s + 2.33·31-s − 1/3·36-s − 1.47·37-s − 0.468·41-s + 0.914·43-s + 0.904·44-s − 0.291·47-s + 49-s − 0.554·52-s − 0.961·53-s + 0.130·59-s − 0.512·61-s − 1.25·63-s − 7/8·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111911 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111911 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.262012356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262012356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 227 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 44 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 118 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.1387745708, −13.7657184365, −13.1584306514, −12.3012554291, −12.0645316820, −11.6533287339, −11.5354992618, −11.0516163061, −10.5127619523, −9.95648199034, −9.32930724597, −9.10245246724, −8.42544754653, −7.98198044021, −7.62628732906, −6.96085007464, −6.60410007760, −5.94077009093, −5.23837547791, −4.85490485456, −4.33073412009, −3.53084020828, −2.74527695298, −1.90160263531, −1.25757098470,
1.25757098470, 1.90160263531, 2.74527695298, 3.53084020828, 4.33073412009, 4.85490485456, 5.23837547791, 5.94077009093, 6.60410007760, 6.96085007464, 7.62628732906, 7.98198044021, 8.42544754653, 9.10245246724, 9.32930724597, 9.95648199034, 10.5127619523, 11.0516163061, 11.5354992618, 11.6533287339, 12.0645316820, 12.3012554291, 13.1584306514, 13.7657184365, 14.1387745708
