| L(s) = 1 | − 2-s − 2·4-s + 2·5-s − 2·7-s + 3·8-s − 2·10-s + 4·11-s + 5·13-s + 2·14-s + 16-s + 17-s − 3·19-s − 4·20-s − 4·22-s + 13·23-s + 3·25-s − 5·26-s + 4·28-s + 29-s − 2·32-s − 34-s − 4·35-s + 3·38-s + 6·40-s + 5·41-s − 2·43-s − 8·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s + 0.894·5-s − 0.755·7-s + 1.06·8-s − 0.632·10-s + 1.20·11-s + 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.688·19-s − 0.894·20-s − 0.852·22-s + 2.71·23-s + 3/5·25-s − 0.980·26-s + 0.755·28-s + 0.185·29-s − 0.353·32-s − 0.171·34-s − 0.676·35-s + 0.486·38-s + 0.948·40-s + 0.780·41-s − 0.304·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560850657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560850657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T - 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 135 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 137 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 85 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 131 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 177 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 137 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 167 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 153 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.04042920231119381932690737441, −9.753766866729512763841275966251, −9.248483680802985052533222341712, −8.959351555927812288297300214900, −8.803396689326762435839335571011, −8.640742040769503528056593481942, −7.70515622230534988845025595180, −7.44588427785290898991507802330, −6.71727343522141888347397982032, −6.44694501343437403443347738583, −6.06620526418433562254995509608, −5.58652582296366593975178811046, −4.84291327957264541803336986666, −4.68200202231709713467618930011, −3.71676745370705737880485753203, −3.70585483573006111685721921414, −2.86809851654240131352188138538, −2.13571733821371376763959408890, −1.03363887182943800339364220791, −0.935176857455447708880149400432,
0.935176857455447708880149400432, 1.03363887182943800339364220791, 2.13571733821371376763959408890, 2.86809851654240131352188138538, 3.70585483573006111685721921414, 3.71676745370705737880485753203, 4.68200202231709713467618930011, 4.84291327957264541803336986666, 5.58652582296366593975178811046, 6.06620526418433562254995509608, 6.44694501343437403443347738583, 6.71727343522141888347397982032, 7.44588427785290898991507802330, 7.70515622230534988845025595180, 8.640742040769503528056593481942, 8.803396689326762435839335571011, 8.959351555927812288297300214900, 9.248483680802985052533222341712, 9.753766866729512763841275966251, 10.04042920231119381932690737441
