| L(s) = 1 | − 2·2-s + 32·4-s + 11·5-s + 259·7-s − 184·8-s − 22·10-s + 269·11-s − 616·13-s − 518·14-s + 368·16-s + 1.89e3·17-s + 164·19-s + 352·20-s − 538·22-s − 3.26e3·23-s + 3.12e3·25-s + 1.23e3·26-s + 8.28e3·28-s − 4.83e3·29-s − 2.84e3·31-s − 5.88e3·32-s − 3.79e3·34-s + 2.84e3·35-s + 1.13e4·37-s − 328·38-s − 2.02e3·40-s + 3.37e4·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 4-s + 0.196·5-s + 1.99·7-s − 1.01·8-s − 0.0695·10-s + 0.670·11-s − 1.01·13-s − 0.706·14-s + 0.359·16-s + 1.59·17-s + 0.104·19-s + 0.196·20-s − 0.236·22-s − 1.28·23-s + 25-s + 0.357·26-s + 1.99·28-s − 1.06·29-s − 0.530·31-s − 1.01·32-s − 0.562·34-s + 0.393·35-s + 1.36·37-s − 0.0368·38-s − 0.200·40-s + 3.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.218706645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.218706645\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 37 p T + p^{5} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T - 7 p^{2} T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 11 T - 3004 T^{2} - 11 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 269 T - 88690 T^{2} - 269 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 308 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1896 T + 2174959 T^{2} - 1896 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 164 T - 2449203 T^{2} - 164 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3264 T + 4217353 T^{2} + 3264 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2417 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2841 T - 20557870 T^{2} + 2841 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 11328 T + 58979627 T^{2} - 11328 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 16856 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7894 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21102 T + 215949397 T^{2} - 21102 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 29691 T + 463359988 T^{2} + 29691 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8163 T - 648289730 T^{2} + 8163 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 15166 T - 614588745 T^{2} + 15166 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 32078 T - 321127023 T^{2} - 32078 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 38274 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 34866 T - 857433637 T^{2} + 34866 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 13529 T - 2894022558 T^{2} + 13529 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 68103 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 114922 T + 7623006635 T^{2} + 114922 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 154959 T + p^{5} T^{2} )^{2} \) |
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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
−14.54348893151507141293156939540, −14.18486076054878851549714353808, −12.95214906774863978430227718158, −12.22124514527243934443952069284, −12.03362765745625803400697044795, −11.33120706085455040414708176564, −11.01689256090200368971508731506, −10.37413804787813767011511561290, −9.434732126130794881203252510319, −9.198792087793825889282597150717, −8.117822211397690101190197204966, −7.69358835229540893068209075687, −7.29647728617382301645682747042, −6.22089359202062142048700507918, −5.64254554922747047630528466922, −4.86049214099348166130608297492, −3.89150614055319611921541105177, −2.60172269683925419811545073202, −1.89117142083973604670015071769, −0.940870785067477514325087956900,
0.940870785067477514325087956900, 1.89117142083973604670015071769, 2.60172269683925419811545073202, 3.89150614055319611921541105177, 4.86049214099348166130608297492, 5.64254554922747047630528466922, 6.22089359202062142048700507918, 7.29647728617382301645682747042, 7.69358835229540893068209075687, 8.117822211397690101190197204966, 9.198792087793825889282597150717, 9.434732126130794881203252510319, 10.37413804787813767011511561290, 11.01689256090200368971508731506, 11.33120706085455040414708176564, 12.03362765745625803400697044795, 12.22124514527243934443952069284, 12.95214906774863978430227718158, 14.18486076054878851549714353808, 14.54348893151507141293156939540
