| L(s) = 1 | − 2·5-s + 3·9-s + 4·19-s + 5·25-s + 4·29-s + 20·31-s − 12·41-s − 6·45-s − 11·49-s − 4·59-s + 18·61-s − 24·71-s + 20·79-s + 9·81-s − 14·89-s − 8·95-s + 30·101-s + 10·109-s + 22·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 9-s + 0.917·19-s + 25-s + 0.742·29-s + 3.59·31-s − 1.87·41-s − 0.894·45-s − 1.57·49-s − 0.520·59-s + 2.30·61-s − 2.84·71-s + 2.25·79-s + 81-s − 1.48·89-s − 0.820·95-s + 2.98·101-s + 0.957·109-s + 2·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.796838626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.796838626\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 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 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.79542562878136897911237145589, −7.57278738022660724743356351083, −7.14198250237981868038764672190, −7.03753563513474298923461755184, −7.03438711846053248958343253254, −6.49713149981109462916226624385, −6.36010710668812766166237250494, −6.21494961412718148333092865764, −6.00543551317566229237836022301, −5.32531175617127739089005443928, −5.31806955794598116055644838702, −4.90811077254735425857358013262, −4.78431362310224122551140672519, −4.52184968061510975186837553528, −4.23909364525897348656444060462, −4.08719133599714794544590945915, −3.52105562549289143233680009821, −3.35437359125743949095575594146, −3.06542520238267471807881358429, −2.82254329471857880622157732076, −2.45006403632303302109154058238, −1.77581895021649955158819762914, −1.64583999910844361337124012831, −0.860766102457155957346083476536, −0.71844865382083135598185067560,
0.71844865382083135598185067560, 0.860766102457155957346083476536, 1.64583999910844361337124012831, 1.77581895021649955158819762914, 2.45006403632303302109154058238, 2.82254329471857880622157732076, 3.06542520238267471807881358429, 3.35437359125743949095575594146, 3.52105562549289143233680009821, 4.08719133599714794544590945915, 4.23909364525897348656444060462, 4.52184968061510975186837553528, 4.78431362310224122551140672519, 4.90811077254735425857358013262, 5.31806955794598116055644838702, 5.32531175617127739089005443928, 6.00543551317566229237836022301, 6.21494961412718148333092865764, 6.36010710668812766166237250494, 6.49713149981109462916226624385, 7.03438711846053248958343253254, 7.03753563513474298923461755184, 7.14198250237981868038764672190, 7.57278738022660724743356351083, 7.79542562878136897911237145589
