| L(s) = 1 | − 130·9-s − 628·11-s − 2.62e3·23-s − 4.13e3·25-s + 1.77e4·29-s + 2.93e3·37-s − 9.83e3·43-s + 1.55e3·53-s − 1.37e4·67-s + 5.66e4·71-s + 2.18e5·79-s − 6.65e4·81-s + 8.16e4·99-s − 3.11e5·107-s + 3.56e5·109-s + 9.35e5·113-s + 3.01e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.64e5·169-s + ⋯ |
| L(s) = 1 | − 0.534·9-s − 1.56·11-s − 1.03·23-s − 1.32·25-s + 3.91·29-s + 0.352·37-s − 0.811·43-s + 0.0758·53-s − 0.372·67-s + 1.33·71-s + 3.93·79-s − 1.12·81-s + 0.837·99-s − 2.63·107-s + 2.87·109-s + 6.88·113-s + 1.87·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 2.05·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.570880867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.570880867\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 130 T^{2} + 9274 p^{2} T^{4} + 130 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4138 T^{2} + 15069186 T^{4} + 4138 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 314 T - 2962 T^{2} + 314 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 764578 T^{2} + 292159956066 T^{4} + 764578 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 691072 T^{2} - 1624510552254 T^{4} + 691072 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 404046 T^{2} + 1324192623914 T^{4} - 404046 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1312 T + 7707614 T^{2} + 1312 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 8866 T + 60324074 T^{2} - 8866 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 92836468 T^{2} + 3695909786961558 T^{4} + 92836468 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 1466 T + 3677778 p T^{2} - 1466 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 276410032 T^{2} + 39064125351165858 T^{4} + 276410032 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 4918 T + 271736814 T^{2} + 4918 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 464693524 T^{2} + 120683872742016534 T^{4} + 464693524 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 776 T + 331555958 T^{2} - 776 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2379976162 T^{2} + 2424748968577420938 T^{4} + 2379976162 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 283151910 T^{2} - 342708036903888286 T^{4} - 283151910 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6852 T + 2598680678 T^{2} + 6852 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 28332 T + 170720206 T^{2} - 28332 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2142879120 T^{2} + 7475057594391926306 T^{4} + 2142879120 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 109148 T + 8896028286 T^{2} - 109148 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 15554590978 T^{2} + 91511105301695799306 T^{4} + 15554590978 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 15004724512 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + 15004724512 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 26479159584 T^{2} + \)\(30\!\cdots\!94\)\( T^{4} + 26479159584 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−6.55639688434563777629948983043, −6.41554858615778695448166007956, −6.10860815460102791154720686251, −5.89113950401273712079225678304, −5.84008824497677940850339044713, −5.30927931000980845870308163202, −5.14582210140878800703956235036, −5.12708023658227288004722454942, −4.56003355390320072460427191071, −4.51256020725086017001100173135, −4.46322247162872994716417353986, −3.84622129825668948234737250928, −3.76957749205374689146933615675, −3.31098903197883027925775373629, −3.06880480942162575780622449277, −2.88098257006900091022679634414, −2.77113310411676300505178965026, −2.16778901206258942046495967187, −2.03591444958387339069535289665, −1.97111588832711553994242208648, −1.53598292274980841790948888096, −0.803771445583756165431061014347, −0.76110422863636012411298765579, −0.59598900651526776494752887366, −0.25921911706266015799840751911,
0.25921911706266015799840751911, 0.59598900651526776494752887366, 0.76110422863636012411298765579, 0.803771445583756165431061014347, 1.53598292274980841790948888096, 1.97111588832711553994242208648, 2.03591444958387339069535289665, 2.16778901206258942046495967187, 2.77113310411676300505178965026, 2.88098257006900091022679634414, 3.06880480942162575780622449277, 3.31098903197883027925775373629, 3.76957749205374689146933615675, 3.84622129825668948234737250928, 4.46322247162872994716417353986, 4.51256020725086017001100173135, 4.56003355390320072460427191071, 5.12708023658227288004722454942, 5.14582210140878800703956235036, 5.30927931000980845870308163202, 5.84008824497677940850339044713, 5.89113950401273712079225678304, 6.10860815460102791154720686251, 6.41554858615778695448166007956, 6.55639688434563777629948983043
