| L(s) = 1 | − 2-s + 3·4-s + 4·5-s − 4·7-s − 4·8-s + 9-s − 4·10-s − 4·11-s − 4·13-s + 4·14-s + 6·16-s − 2·17-s − 18-s − 4·19-s + 12·20-s + 4·22-s − 12·23-s + 10·25-s + 4·26-s − 12·28-s + 6·29-s − 6·32-s + 2·34-s − 16·35-s + 3·36-s − 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3/2·4-s + 1.78·5-s − 1.51·7-s − 1.41·8-s + 1/3·9-s − 1.26·10-s − 1.20·11-s − 1.10·13-s + 1.06·14-s + 3/2·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 2.68·20-s + 0.852·22-s − 2.50·23-s + 2·25-s + 0.784·26-s − 2.26·28-s + 1.11·29-s − 1.06·32-s + 0.342·34-s − 2.70·35-s + 1/2·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5898977458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5898977458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - p T^{2} - T^{3} + 3 T^{4} - p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 4 T^{3} + 8 T^{4} - 4 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 5 T^{2} - 4 T^{3} + 144 T^{4} - 4 p T^{5} - 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 11 T^{2} - 38 T^{3} - 132 T^{4} - 38 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 21 T^{2} - 4 T^{3} + 704 T^{4} - 4 p T^{5} - 21 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 67 T^{2} + 372 T^{3} + 2088 T^{4} + 372 p T^{5} + 67 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 11 T^{2} + 66 T^{3} + 324 T^{4} + 66 p T^{5} - 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} - 426 T^{3} - 3036 T^{4} - 426 p T^{5} + 25 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 33 T^{2} - 88 T^{3} + 4808 T^{4} - 88 p T^{5} - 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 35 T^{2} - 108 T^{3} - 96 T^{4} - 108 p T^{5} + 35 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 41 T^{2} - 232 T^{3} + 2248 T^{4} - 232 p T^{5} - 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T - 89 T^{2} - 88 T^{3} + 13824 T^{4} - 88 p T^{5} - 89 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 171 T^{2} - 38 T^{3} + 20828 T^{4} - 38 p T^{5} - 171 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.89200202524759998930627850597, −10.42568381892046713289286935250, −10.36036668894980585211914562316, −10.30938188485024706140476848320, −10.01009925698353961301550733886, −9.884308230922131675060388385853, −9.229375035040672833821934710016, −9.015218288918786119772358897591, −8.813569820285216209934805029874, −8.501118814250077106126849832631, −7.84312595428948964394421230079, −7.58676772867519282729109863211, −7.14178766249817827501905798296, −7.07440738516544523845869778812, −6.37010337587138669427635071090, −6.27573316742424405601050745681, −6.09431055068922998204347764681, −5.53560787487605899639312025354, −5.40477671216775047552365708067, −4.59101232351657401245294500722, −4.11077043036065636618792526652, −3.37901253656634988939893715786, −2.51624338172632343298722397469, −2.50969198149248296814249085003, −2.07564039327805673978410564140,
2.07564039327805673978410564140, 2.50969198149248296814249085003, 2.51624338172632343298722397469, 3.37901253656634988939893715786, 4.11077043036065636618792526652, 4.59101232351657401245294500722, 5.40477671216775047552365708067, 5.53560787487605899639312025354, 6.09431055068922998204347764681, 6.27573316742424405601050745681, 6.37010337587138669427635071090, 7.07440738516544523845869778812, 7.14178766249817827501905798296, 7.58676772867519282729109863211, 7.84312595428948964394421230079, 8.501118814250077106126849832631, 8.813569820285216209934805029874, 9.015218288918786119772358897591, 9.229375035040672833821934710016, 9.884308230922131675060388385853, 10.01009925698353961301550733886, 10.30938188485024706140476848320, 10.36036668894980585211914562316, 10.42568381892046713289286935250, 10.89200202524759998930627850597
