| L(s) = 1 | − 3·2-s − 2·3-s + 3·4-s − 3·5-s + 6·6-s + 9-s + 9·10-s − 6·12-s − 14·13-s + 6·15-s − 2·16-s + 6·17-s − 3·18-s − 9·20-s − 6·23-s − 3·25-s + 42·26-s − 4·27-s + 9·29-s − 18·30-s − 30·31-s − 6·32-s − 18·34-s + 3·36-s − 24·37-s + 28·39-s − 18·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 2.44·6-s + 1/3·9-s + 2.84·10-s − 1.73·12-s − 3.88·13-s + 1.54·15-s − 1/2·16-s + 1.45·17-s − 0.707·18-s − 2.01·20-s − 1.25·23-s − 3/5·25-s + 8.23·26-s − 0.769·27-s + 1.67·29-s − 3.28·30-s − 5.38·31-s − 1.06·32-s − 3.08·34-s + 1/2·36-s − 3.94·37-s + 4.48·39-s − 2.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 27 T^{3} + 71 T^{4} + 27 p T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 27 T^{2} + 377 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 537 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 210 T^{2} + 1836 T^{3} + 13151 T^{4} + 1836 p T^{5} + 210 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 24 T + 327 T^{2} + 3240 T^{3} + 25040 T^{4} + 3240 p T^{5} + 327 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 21 T^{2} - 54 T^{3} - 2692 T^{4} - 54 p T^{5} + 21 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 33 T + 588 T^{2} + 7425 T^{3} + 75011 T^{4} + 7425 p T^{5} + 588 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 39 T + 812 T^{2} - 11895 T^{3} + 132795 T^{4} - 11895 p T^{5} + 812 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.303402142257381408112723300160, −7.82587392553078210127500271384, −7.64677250007074815033225172550, −7.25343191510540273514513274793, −7.25024355995263600117105028241, −7.18905558440256622046945218427, −6.95645538376390165431511026074, −6.80135741074868912472326485978, −6.17124881041506771497473506334, −6.04682563206607060888512777965, −5.45459392373462712278001370540, −5.40875844360904347246112804037, −5.21519282429262463344276603999, −5.00027443910550228092071050657, −4.98262095179010938913852051827, −4.62374025012526072344057157228, −3.83316924665352311824692119410, −3.72838439381942006519464339291, −3.63688282479232588270682973216, −3.43820993912466889096361815679, −3.00238638961305347992632021906, −2.19183216262703043005804863449, −1.96800491764636992644598015215, −1.90064063071080855486757651577, −1.38228298275911195473528000535, 0, 0, 0, 0,
1.38228298275911195473528000535, 1.90064063071080855486757651577, 1.96800491764636992644598015215, 2.19183216262703043005804863449, 3.00238638961305347992632021906, 3.43820993912466889096361815679, 3.63688282479232588270682973216, 3.72838439381942006519464339291, 3.83316924665352311824692119410, 4.62374025012526072344057157228, 4.98262095179010938913852051827, 5.00027443910550228092071050657, 5.21519282429262463344276603999, 5.40875844360904347246112804037, 5.45459392373462712278001370540, 6.04682563206607060888512777965, 6.17124881041506771497473506334, 6.80135741074868912472326485978, 6.95645538376390165431511026074, 7.18905558440256622046945218427, 7.25024355995263600117105028241, 7.25343191510540273514513274793, 7.64677250007074815033225172550, 7.82587392553078210127500271384, 8.303402142257381408112723300160
