| L(s) = 1 | − 3-s + 4·5-s − 3·7-s − 9-s − 2·11-s − 3·13-s − 4·15-s + 3·17-s + 2·19-s + 3·21-s − 7·23-s + 2·25-s + 29-s − 4·31-s + 2·33-s − 12·35-s − 2·37-s + 3·39-s + 4·41-s − 4·45-s − 16·47-s − 3·49-s − 3·51-s + 5·53-s − 8·55-s − 2·57-s + 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.13·7-s − 1/3·9-s − 0.603·11-s − 0.832·13-s − 1.03·15-s + 0.727·17-s + 0.458·19-s + 0.654·21-s − 1.45·23-s + 2/5·25-s + 0.185·29-s − 0.718·31-s + 0.348·33-s − 2.02·35-s − 0.328·37-s + 0.480·39-s + 0.624·41-s − 0.596·45-s − 2.33·47-s − 3/7·49-s − 0.420·51-s + 0.686·53-s − 1.07·55-s − 0.264·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 98 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.241358901127949633502169989892, −8.168938202621253209161008188640, −7.70449044677278006478136073664, −7.24090246049803484143716296167, −6.68660139565452404036237251892, −6.62556465392806951888721575258, −6.01836922646444643217790656664, −5.72154990308849569754662416385, −5.51584129003431748766826359571, −5.37435559285531567292872946628, −4.66891664935232374875785439530, −4.22657008954923117030858530855, −3.65445867279053163918845891013, −3.10985526071700341300503710576, −2.78701955004407696067589368274, −2.29194267295121904310005997608, −1.75470755572988585628446622639, −1.34577154348212522773253920612, 0, 0,
1.34577154348212522773253920612, 1.75470755572988585628446622639, 2.29194267295121904310005997608, 2.78701955004407696067589368274, 3.10985526071700341300503710576, 3.65445867279053163918845891013, 4.22657008954923117030858530855, 4.66891664935232374875785439530, 5.37435559285531567292872946628, 5.51584129003431748766826359571, 5.72154990308849569754662416385, 6.01836922646444643217790656664, 6.62556465392806951888721575258, 6.68660139565452404036237251892, 7.24090246049803484143716296167, 7.70449044677278006478136073664, 8.168938202621253209161008188640, 8.241358901127949633502169989892
