L(s) = 1 | − 2-s + 2·4-s − 2·5-s − 5·8-s + 2·10-s + 4·11-s − 4·13-s + 5·16-s − 6·17-s − 4·19-s − 4·20-s − 4·22-s + 5·25-s + 4·26-s + 4·29-s − 10·32-s + 6·34-s − 6·37-s + 4·38-s + 10·40-s − 4·41-s − 8·43-s + 8·44-s − 5·50-s − 8·52-s + 6·53-s − 8·55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 4-s − 0.894·5-s − 1.76·8-s + 0.632·10-s + 1.20·11-s − 1.10·13-s + 5/4·16-s − 1.45·17-s − 0.917·19-s − 0.894·20-s − 0.852·22-s + 25-s + 0.784·26-s + 0.742·29-s − 1.76·32-s + 1.02·34-s − 0.986·37-s + 0.648·38-s + 1.58·40-s − 0.624·41-s − 1.21·43-s + 1.20·44-s − 0.707·50-s − 1.10·52-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6955965604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6955965604\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−11.80402897285165985849555844608, −10.81816057609690301350209065938, −10.61409736355128903825277258672, −9.863927074629101212979322956917, −9.613997248842167876723362570465, −8.785828459576528146697069600297, −8.736676061247533233536899379302, −8.448891794333314692240576878529, −7.63060349210109494651241503445, −7.05056013115321086583392089416, −6.85542351953130888969222340940, −6.38833289892065249201816713922, −5.98548577948485881999583093687, −4.82861947041117472738471645570, −4.79208794793551421877151840854, −3.56093595562485365691255699573, −3.55528996331510143687911431437, −2.38234167632467782041601104852, −2.06372222323479133124052619410, −0.56574615748523627800600037904,
0.56574615748523627800600037904, 2.06372222323479133124052619410, 2.38234167632467782041601104852, 3.55528996331510143687911431437, 3.56093595562485365691255699573, 4.79208794793551421877151840854, 4.82861947041117472738471645570, 5.98548577948485881999583093687, 6.38833289892065249201816713922, 6.85542351953130888969222340940, 7.05056013115321086583392089416, 7.63060349210109494651241503445, 8.448891794333314692240576878529, 8.736676061247533233536899379302, 8.785828459576528146697069600297, 9.613997248842167876723362570465, 9.863927074629101212979322956917, 10.61409736355128903825277258672, 10.81816057609690301350209065938, 11.80402897285165985849555844608