| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 4·7-s − 4·8-s + 4·10-s + 5·11-s − 6·13-s − 8·14-s + 5·16-s − 4·17-s + 4·19-s − 6·20-s − 10·22-s − 4·23-s + 5·25-s + 12·26-s + 12·28-s + 7·29-s + 6·31-s − 6·32-s + 8·34-s − 8·35-s − 8·37-s − 8·38-s + 8·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.51·7-s − 1.41·8-s + 1.26·10-s + 1.50·11-s − 1.66·13-s − 2.13·14-s + 5/4·16-s − 0.970·17-s + 0.917·19-s − 1.34·20-s − 2.13·22-s − 0.834·23-s + 25-s + 2.35·26-s + 2.26·28-s + 1.29·29-s + 1.07·31-s − 1.06·32-s + 1.37·34-s − 1.35·35-s − 1.31·37-s − 1.29·38-s + 1.26·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7930216919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7930216919\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.05400919421379438760999430116, −9.420534080915034456056269298063, −9.333973330882336132479601454115, −8.645331910880944980061962561194, −8.247289127717729374326865319665, −8.192113434779701611936523038023, −7.77836407069395529231588633577, −7.02396428498687281853860832898, −6.99232794697336299201483289464, −6.58683886370695616937023373740, −6.03330891931902211325166598718, −5.03594169533755216782410704213, −4.96090209721586719185369322622, −4.48802954060018139121540045589, −3.84176243225112107193779789095, −3.15765436029107410210193812541, −2.65408615072953286908197939563, −1.64865846900796375512053660548, −1.63206705601927350314656240401, −0.50455770105227996056016793511,
0.50455770105227996056016793511, 1.63206705601927350314656240401, 1.64865846900796375512053660548, 2.65408615072953286908197939563, 3.15765436029107410210193812541, 3.84176243225112107193779789095, 4.48802954060018139121540045589, 4.96090209721586719185369322622, 5.03594169533755216782410704213, 6.03330891931902211325166598718, 6.58683886370695616937023373740, 6.99232794697336299201483289464, 7.02396428498687281853860832898, 7.77836407069395529231588633577, 8.192113434779701611936523038023, 8.247289127717729374326865319665, 8.645331910880944980061962561194, 9.333973330882336132479601454115, 9.420534080915034456056269298063, 10.05400919421379438760999430116
