| L(s) = 1 | + 250·25-s − 1.22e3·29-s − 478·81-s + 5.54e3·89-s − 1.51e3·101-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 2·25-s − 7.83·29-s − 0.655·81-s + 6.60·89-s − 1.48·101-s − 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5088585655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5088585655\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 134 T + 8978 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} )( 1 + 134 T + 8978 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + 306 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 74338 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 594 T + 176418 T^{2} - 594 p^{3} T^{3} + p^{6} T^{4} )( 1 + 594 T + 176418 T^{2} + 594 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 602 T + 181202 T^{2} - 602 p^{3} T^{3} + p^{6} T^{4} )( 1 + 602 T + 181202 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 452342 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1098 T + 602802 T^{2} - 1098 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1098 T + 602802 T^{2} + 1098 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 154 T + 11858 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} )( 1 + 154 T + 11858 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 1386 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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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
−9.142440889879378783652296202863, −9.084321433929143711189704304566, −8.296471432672225401798581744118, −8.121616736501044980628185671036, −7.76447838998563013933559703178, −7.52858319916217016773528016669, −7.42208168823084292373623505920, −6.98578195471648706148523551539, −6.86068672533443588908611754775, −6.38507898824676997381134157602, −5.90699923915792669497212227410, −5.89157886146161490282339699735, −5.35982987533224370712558748595, −5.13166426923616119496647256907, −5.10593770879553400022617563211, −4.32361141496976920208105231329, −4.05964249130542937344421446193, −3.59059949057182386486378014440, −3.55265180026937262639531173105, −3.08780382429683217382433433786, −2.36667406218455127757591975495, −1.93666941953532595376525967970, −1.78298412618732499995550685063, −0.997949833365966833715204881377, −0.16759679735773697081094993538,
0.16759679735773697081094993538, 0.997949833365966833715204881377, 1.78298412618732499995550685063, 1.93666941953532595376525967970, 2.36667406218455127757591975495, 3.08780382429683217382433433786, 3.55265180026937262639531173105, 3.59059949057182386486378014440, 4.05964249130542937344421446193, 4.32361141496976920208105231329, 5.10593770879553400022617563211, 5.13166426923616119496647256907, 5.35982987533224370712558748595, 5.89157886146161490282339699735, 5.90699923915792669497212227410, 6.38507898824676997381134157602, 6.86068672533443588908611754775, 6.98578195471648706148523551539, 7.42208168823084292373623505920, 7.52858319916217016773528016669, 7.76447838998563013933559703178, 8.121616736501044980628185671036, 8.296471432672225401798581744118, 9.084321433929143711189704304566, 9.142440889879378783652296202863
