| L(s) = 1 | + 4.28e3·5-s + 1.28e5·13-s + 3.86e5·17-s + 6.58e6·25-s − 1.24e6·29-s + 5.30e7·37-s − 1.63e7·41-s − 1.00e7·49-s + 1.43e8·53-s + 7.19e7·61-s + 5.51e8·65-s − 2.54e8·73-s + 1.65e9·85-s − 8.10e8·89-s − 2.46e9·97-s + 5.07e9·101-s + 1.29e9·109-s − 5.11e8·113-s − 1.82e9·121-s + 1.43e9·125-s + ⋯ |
| L(s) = 1 | + 3.06·5-s + 1.24·13-s + 1.12·17-s + 3.37·25-s − 0.326·29-s + 4.65·37-s − 0.904·41-s − 0.249·49-s + 2.49·53-s + 0.665·61-s + 3.83·65-s − 1.05·73-s + 3.44·85-s − 1.36·89-s − 2.82·97-s + 4.84·101-s + 0.881·109-s − 0.295·113-s − 0.772·121-s + 0.526·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(38.68375679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(38.68375679\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2144 T + 720754 p T^{2} - 2144 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 1436164 p T^{2} + 1493884390218 p^{3} T^{4} + 1436164 p^{19} T^{6} + p^{36} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 1821033644 T^{2} + 11917965741885833046 T^{4} + 1821033644 p^{18} T^{6} + p^{36} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 64308 T + 12952228862 T^{2} - 64308 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 193216 T + 68680018658 T^{2} - 193216 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 405943332404 T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - 405943332404 p^{18} T^{6} + p^{36} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1259879112868 T^{2} + \)\(56\!\cdots\!94\)\( T^{4} - 1259879112868 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 21472 p T + 21196428285674 T^{2} + 21472 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 2973858190084 p T^{2} + \)\(35\!\cdots\!86\)\( T^{4} + 2973858190084 p^{19} T^{6} + p^{36} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 26510396 T + 405438433058958 T^{2} - 26510396 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 8180416 T + 666308181296786 T^{2} + 8180416 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 90916615908652 T^{2} - \)\(28\!\cdots\!26\)\( T^{4} + 90916615908652 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 2859901033920188 T^{2} + \)\(45\!\cdots\!14\)\( T^{4} + 2859901033920188 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 71553952 T + 7725362603865242 T^{2} - 71553952 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 5666480964301676 T^{2} + \)\(13\!\cdots\!86\)\( T^{4} + 5666480964301676 p^{18} T^{6} + p^{36} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 35966380 T + 17202946208621982 T^{2} - 35966380 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 13907212634221708 T^{2} + \)\(76\!\cdots\!34\)\( T^{4} + 13907212634221708 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 110718934748939804 T^{2} + \)\(66\!\cdots\!26\)\( T^{4} + 110718934748939804 p^{18} T^{6} + p^{36} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 127397748 T + 16788098748021302 T^{2} + 127397748 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 216241825033102396 T^{2} + \)\(28\!\cdots\!26\)\( T^{4} + 216241825033102396 p^{18} T^{6} + p^{36} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 215613462891120332 T^{2} + \)\(13\!\cdots\!74\)\( T^{4} + 215613462891120332 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 405455488 T - 66712792477860046 T^{2} + 405455488 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1233630756 T + 1796783470252454918 T^{2} + 1233630756 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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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
−7.07505521081527820964327455504, −6.29275139100412383284807207723, −6.21302018903623144422866167771, −6.20868333822278152766284572252, −6.18670520147261651328007668647, −5.63502710493443819880626352025, −5.54839353589893128827698142974, −5.31468121480826932292387018984, −5.17140276524811046115781538099, −4.57093091233265918693113670858, −4.24060310580228063198851492116, −4.22263954644733678441750035362, −3.78702292894079524053657672578, −3.44307751862576917680118643351, −3.04979182290433301228643614508, −2.77405175616684663363837209529, −2.55704693411507340872512550457, −2.34675217411512140983451801919, −1.82477341072574314720665507655, −1.69214022415124943770212441671, −1.66432185529370340918250680977, −1.16092890276060678061481175859, −0.73545426942465641832134288784, −0.69405473807598144589422772995, −0.47688297527187534168054024083,
0.47688297527187534168054024083, 0.69405473807598144589422772995, 0.73545426942465641832134288784, 1.16092890276060678061481175859, 1.66432185529370340918250680977, 1.69214022415124943770212441671, 1.82477341072574314720665507655, 2.34675217411512140983451801919, 2.55704693411507340872512550457, 2.77405175616684663363837209529, 3.04979182290433301228643614508, 3.44307751862576917680118643351, 3.78702292894079524053657672578, 4.22263954644733678441750035362, 4.24060310580228063198851492116, 4.57093091233265918693113670858, 5.17140276524811046115781538099, 5.31468121480826932292387018984, 5.54839353589893128827698142974, 5.63502710493443819880626352025, 6.18670520147261651328007668647, 6.20868333822278152766284572252, 6.21302018903623144422866167771, 6.29275139100412383284807207723, 7.07505521081527820964327455504
