L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 6·5-s + 8·6-s − 4·7-s + 2·8-s + 7·9-s + 12·10-s − 7·11-s − 4·12-s − 7·13-s + 8·14-s + 24·15-s − 3·16-s − 17-s − 14·18-s − 4·19-s − 6·20-s + 16·21-s + 14·22-s − 2·23-s − 8·24-s + 19·25-s + 14·26-s − 4·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.68·5-s + 3.26·6-s − 1.51·7-s + 0.707·8-s + 7/3·9-s + 3.79·10-s − 2.11·11-s − 1.15·12-s − 1.94·13-s + 2.13·14-s + 6.19·15-s − 3/4·16-s − 0.242·17-s − 3.29·18-s − 0.917·19-s − 1.34·20-s + 3.49·21-s + 2.98·22-s − 0.417·23-s − 1.63·24-s + 19/5·25-s + 2.74·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46234 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46234 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 23117 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 200 T + p T^{2} ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 53 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 4 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 22 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−15.6793869195, −15.5168826626, −15.0393091864, −14.3134051489, −13.3687166354, −12.6583702437, −12.5417590300, −12.2795026088, −11.7695952259, −11.2250233830, −10.7737929161, −10.5085643553, −10.1883802410, −9.52593292253, −8.82037515270, −8.28873035344, −7.61478704025, −7.40141515679, −7.06163721341, −6.25901498483, −5.47971240670, −4.97858294024, −4.47601985660, −3.67112207875, −2.72297767378, 0, 0, 0,
2.72297767378, 3.67112207875, 4.47601985660, 4.97858294024, 5.47971240670, 6.25901498483, 7.06163721341, 7.40141515679, 7.61478704025, 8.28873035344, 8.82037515270, 9.52593292253, 10.1883802410, 10.5085643553, 10.7737929161, 11.2250233830, 11.7695952259, 12.2795026088, 12.5417590300, 12.6583702437, 13.3687166354, 14.3134051489, 15.0393091864, 15.5168826626, 15.6793869195