| L(s) = 1 | + 4-s + 3·9-s + 4·11-s − 12·19-s − 36·29-s + 8·31-s + 3·36-s − 28·41-s + 4·44-s + 13·49-s + 20·59-s − 2·61-s − 64-s + 8·71-s − 12·76-s + 20·79-s + 9·81-s + 2·89-s + 12·99-s − 6·101-s − 18·109-s − 36·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 9-s + 1.20·11-s − 2.75·19-s − 6.68·29-s + 1.43·31-s + 1/2·36-s − 4.37·41-s + 0.603·44-s + 13/7·49-s + 2.60·59-s − 0.256·61-s − 1/8·64-s + 0.949·71-s − 1.37·76-s + 2.25·79-s + 81-s + 0.211·89-s + 1.20·99-s − 0.597·101-s − 1.72·109-s − 3.34·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666786501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666786501\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 102 T^{2} + 7595 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.212493940436230038301393131032, −8.180185161112396790357306212199, −7.978059733327995750764723036938, −7.51263425677502025768590569392, −7.17785276265171945593343973988, −6.91775225261044768116889079669, −6.90954733835177592113914769354, −6.65823937061205678697031699098, −6.54313216531256782082829026907, −5.86910726986214489716337011705, −5.83062831076453301823990440856, −5.49303654544934266054455343763, −5.37638654993758250888226586360, −4.81872899830111916020011409958, −4.60640276763200156581279358705, −3.99346796826565650026334032043, −3.97614409054278345314546042821, −3.90683987712315382891813101600, −3.38883812772728956687879709510, −3.22269022082109992907394517546, −2.29224105119528980137942585785, −1.96389021575878599347703612669, −1.93660913089026135260702039614, −1.65367680264202821369297322985, −0.50550413380204383980470809741,
0.50550413380204383980470809741, 1.65367680264202821369297322985, 1.93660913089026135260702039614, 1.96389021575878599347703612669, 2.29224105119528980137942585785, 3.22269022082109992907394517546, 3.38883812772728956687879709510, 3.90683987712315382891813101600, 3.97614409054278345314546042821, 3.99346796826565650026334032043, 4.60640276763200156581279358705, 4.81872899830111916020011409958, 5.37638654993758250888226586360, 5.49303654544934266054455343763, 5.83062831076453301823990440856, 5.86910726986214489716337011705, 6.54313216531256782082829026907, 6.65823937061205678697031699098, 6.90954733835177592113914769354, 6.91775225261044768116889079669, 7.17785276265171945593343973988, 7.51263425677502025768590569392, 7.978059733327995750764723036938, 8.180185161112396790357306212199, 8.212493940436230038301393131032
