L(s) = 1 | + 6·9-s + 32·11-s − 8·17-s − 32·19-s + 28·25-s + 40·41-s − 32·43-s + 76·49-s + 128·59-s + 256·67-s + 200·73-s + 27·81-s − 160·83-s − 200·89-s + 56·97-s + 192·99-s + 320·107-s − 344·113-s + 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + ⋯ |
L(s) = 1 | + 2/3·9-s + 2.90·11-s − 0.470·17-s − 1.68·19-s + 1.11·25-s + 0.975·41-s − 0.744·43-s + 1.55·49-s + 2.16·59-s + 3.82·67-s + 2.73·73-s + 1/3·81-s − 1.92·83-s − 2.24·89-s + 0.577·97-s + 1.93·99-s + 2.99·107-s − 3.04·113-s + 1.28·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.313·153-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.683083744\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.683083744\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 5 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 390 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 738 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 16 T + 3330 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 64 T + 7554 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 128 T + 12642 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 80 T + 14610 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−10.07658144442517021937209412550, −9.650031991667628483914487083457, −9.525964134755084538091811670113, −9.193307244819598524491510999011, −8.984623839582391828825584813326, −8.629283470338112893205699526480, −8.353235085482889908631035353596, −8.201039569535917050627134093300, −7.72527875843249215604856316257, −7.09762586723807530275583481679, −6.81659379983574007725336817223, −6.81294755735736500508846832778, −6.64185458853050468309583150079, −6.02012144720189482322314123805, −5.95283299149645828436300591544, −5.10212536718697014132861994491, −5.06533523333496423088131741856, −4.44289649370149009860206090090, −3.89574298449500398082031596806, −3.83767617505855486641527890487, −3.76758436922444267332777499929, −2.52380985762095774734926340682, −2.35503825140727610443945900475, −1.46272874626522415542166720358, −0.929413234716795791901505677024,
0.929413234716795791901505677024, 1.46272874626522415542166720358, 2.35503825140727610443945900475, 2.52380985762095774734926340682, 3.76758436922444267332777499929, 3.83767617505855486641527890487, 3.89574298449500398082031596806, 4.44289649370149009860206090090, 5.06533523333496423088131741856, 5.10212536718697014132861994491, 5.95283299149645828436300591544, 6.02012144720189482322314123805, 6.64185458853050468309583150079, 6.81294755735736500508846832778, 6.81659379983574007725336817223, 7.09762586723807530275583481679, 7.72527875843249215604856316257, 8.201039569535917050627134093300, 8.353235085482889908631035353596, 8.629283470338112893205699526480, 8.984623839582391828825584813326, 9.193307244819598524491510999011, 9.525964134755084538091811670113, 9.650031991667628483914487083457, 10.07658144442517021937209412550