| L(s) = 1 | + 150·5-s + 3.33e3·9-s + 1.36e4·11-s − 1.37e4·19-s − 5.56e4·25-s + 5.11e4·29-s − 1.64e5·31-s − 1.06e6·41-s + 4.99e5·45-s + 1.47e6·49-s + 2.04e6·55-s − 2.87e6·59-s + 2.76e6·61-s + 9.63e5·71-s + 2.11e6·79-s + 6.30e6·81-s + 1.12e7·89-s − 2.05e6·95-s + 4.54e7·99-s + 1.02e7·101-s − 4.02e7·109-s + 1.00e8·121-s − 2.00e7·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 1.52·9-s + 3.09·11-s − 0.458·19-s − 0.711·25-s + 0.389·29-s − 0.990·31-s − 2.41·41-s + 0.817·45-s + 1.78·49-s + 1.66·55-s − 1.82·59-s + 1.55·61-s + 0.319·71-s + 0.483·79-s + 1.31·81-s + 1.69·89-s − 0.246·95-s + 4.71·99-s + 0.993·101-s − 2.97·109-s + 5.17·121-s − 0.918·125-s + 0.209·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.689756955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.689756955\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 p^{2} T + p^{7} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 370 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1470650 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6828 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22562810 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 574764770 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6860 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5955848090 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 25590 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 82112 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 139899246410 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 533118 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 41047812050 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1013212289930 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2002060594730 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 1438980 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1381022 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 4750924642370 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 481608 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 19886077213490 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1059760 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 47492314121570 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5644170 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17378330046530 T^{2} + p^{14} T^{4} \) |
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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
−13.60553350606698602573653612800, −12.58577102642512439101818671662, −11.98780745684260189955008652755, −11.89612578746256418179398706115, −11.03443752249341754920623295135, −10.33280104531768416665887642252, −9.866889573682742400097092268080, −9.173451997319371395086381194364, −9.081570993948299991404402892966, −8.150691049333939395205349107648, −7.21761053743527145950407970145, −6.70663129341328057057407092571, −6.43686840948755418950236743948, −5.54099651426591405904584665354, −4.52637263393600351171371996973, −3.98373212408078119253774394748, −3.49958255735858424666960578147, −1.89503017927863097270665853175, −1.59789182318310985923922588707, −0.78210526767627408007786715253,
0.78210526767627408007786715253, 1.59789182318310985923922588707, 1.89503017927863097270665853175, 3.49958255735858424666960578147, 3.98373212408078119253774394748, 4.52637263393600351171371996973, 5.54099651426591405904584665354, 6.43686840948755418950236743948, 6.70663129341328057057407092571, 7.21761053743527145950407970145, 8.150691049333939395205349107648, 9.081570993948299991404402892966, 9.173451997319371395086381194364, 9.866889573682742400097092268080, 10.33280104531768416665887642252, 11.03443752249341754920623295135, 11.89612578746256418179398706115, 11.98780745684260189955008652755, 12.58577102642512439101818671662, 13.60553350606698602573653612800
