L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 5·7-s + 4·8-s + 3·9-s + 4·10-s − 6·12-s − 3·13-s − 10·14-s − 4·15-s + 5·16-s + 4·17-s + 6·18-s − 3·19-s + 6·20-s + 10·21-s − 15·23-s − 8·24-s + 3·25-s − 6·26-s − 4·27-s − 15·28-s + 10·29-s − 8·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.88·7-s + 1.41·8-s + 9-s + 1.26·10-s − 1.73·12-s − 0.832·13-s − 2.67·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 0.688·19-s + 1.34·20-s + 2.18·21-s − 3.12·23-s − 1.63·24-s + 3/5·25-s − 1.17·26-s − 0.769·27-s − 2.83·28-s + 1.85·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
good | 7 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 15 T + 101 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 105 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - 57 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 131 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.099539965585419225973610808972, −7.87065914553810917024590194379, −7.36408771339446988748830466419, −7.00065817729785102810165062017, −6.57276791632986721839295974336, −6.35587051191708998860383390160, −5.99268462097586925552462769469, −5.90747957522273581367819695450, −5.33159486445693684658443115418, −5.16274164365577208151797873229, −4.46985032305551930992093064408, −4.28079696463934018877004978494, −3.55841137571843737698379049480, −3.53365866370484679943565569113, −2.68376304686716637507400671629, −2.58478702253765011863624440915, −1.62623074245142524409813029989, −1.55769799043235622931016301067, 0, 0,
1.55769799043235622931016301067, 1.62623074245142524409813029989, 2.58478702253765011863624440915, 2.68376304686716637507400671629, 3.53365866370484679943565569113, 3.55841137571843737698379049480, 4.28079696463934018877004978494, 4.46985032305551930992093064408, 5.16274164365577208151797873229, 5.33159486445693684658443115418, 5.90747957522273581367819695450, 5.99268462097586925552462769469, 6.35587051191708998860383390160, 6.57276791632986721839295974336, 7.00065817729785102810165062017, 7.36408771339446988748830466419, 7.87065914553810917024590194379, 8.099539965585419225973610808972