| 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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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
−7.68189823196117294097268548935, −7.44848366539483132526658859290, −6.93169299753632751994671263187, −6.82516929185586179634358724223, −6.52533629479449240434099143650, −6.14246715518054272223203186373, −6.05753961440893266118204520488, −5.83417611911358836006049493595, −5.46187563990275587647989093321, −4.83910370959326689657540195848, −4.76338096906034482169327299793, −4.44532228836936230377039238572, −4.38647199001776449069375663527, −3.95996312113978344113416966272, −3.77482870513403635756892747702, −3.66758878656067014252097733536, −3.46819550238916174612050607963, −2.83184794331045936504105971394, −2.62891474163835539362944976745, −2.50849508964692756800472237852, −2.21742213111656386292378289641, −1.42202247383701842112579760629, −1.17110907306671215106091977015, −1.12281461376311054741531167812, −0.885034429184231961454259108924, 0, 0, 0, 0,
0.885034429184231961454259108924, 1.12281461376311054741531167812, 1.17110907306671215106091977015, 1.42202247383701842112579760629, 2.21742213111656386292378289641, 2.50849508964692756800472237852, 2.62891474163835539362944976745, 2.83184794331045936504105971394, 3.46819550238916174612050607963, 3.66758878656067014252097733536, 3.77482870513403635756892747702, 3.95996312113978344113416966272, 4.38647199001776449069375663527, 4.44532228836936230377039238572, 4.76338096906034482169327299793, 4.83910370959326689657540195848, 5.46187563990275587647989093321, 5.83417611911358836006049493595, 6.05753961440893266118204520488, 6.14246715518054272223203186373, 6.52533629479449240434099143650, 6.82516929185586179634358724223, 6.93169299753632751994671263187, 7.44848366539483132526658859290, 7.68189823196117294097268548935
