| L(s) = 1 | − 1.37e3·7-s + 1.32e4·13-s − 9.18e3·19-s − 2.39e4·25-s − 4.45e5·31-s − 8.52e5·37-s + 1.50e5·43-s + 1.17e6·49-s − 1.30e6·61-s − 6.07e6·67-s + 3.63e6·73-s − 1.68e7·79-s − 1.82e7·91-s − 3.64e7·97-s − 3.09e7·103-s − 3.19e7·109-s − 6.66e7·121-s + 127-s + 131-s + 1.26e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.67·13-s − 0.307·19-s − 0.307·25-s − 2.68·31-s − 2.76·37-s + 0.288·43-s + 10/7·49-s − 0.734·61-s − 2.46·67-s + 1.09·73-s − 3.85·79-s − 2.53·91-s − 4.05·97-s − 2.78·103-s − 2.36·109-s − 3.41·121-s + 0.464·133-s − 2.13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 23988 T^{2} - 337917994 p^{2} T^{4} + 23988 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 66638700 T^{2} + 1869640076065382 T^{4} + 66638700 p^{14} T^{6} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 6636 T + 132998174 T^{2} - 6636 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 47340540 T^{2} + 304481267672814902 T^{4} - 47340540 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4592 T + 779207718 T^{2} + 4592 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1458162852 T^{2} + 22257185194263703190 T^{4} - 1458162852 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 38888338836 T^{2} + \)\(77\!\cdots\!62\)\( T^{4} + 38888338836 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 222768 T + 37091070062 T^{2} + 222768 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 426404 T + 127886837070 T^{2} + 426404 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 20538923492 T^{2} - \)\(33\!\cdots\!18\)\( T^{4} + 20538923492 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 75112 T - 159018745386 T^{2} - 75112 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 1246374912572 T^{2} + \)\(89\!\cdots\!50\)\( T^{4} + 1246374912572 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1583203178380 T^{2} + \)\(33\!\cdots\!42\)\( T^{4} - 1583203178380 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 4638308305356 T^{2} + \)\(10\!\cdots\!22\)\( T^{4} + 4638308305356 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 651420 T + 6011864979998 T^{2} + 651420 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 3035848 T + 13583249522022 T^{2} + 3035848 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 18965138502300 T^{2} + \)\(18\!\cdots\!62\)\( T^{4} + 18965138502300 p^{14} T^{6} + p^{28} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1815884 T + 3480307846614 T^{2} - 1815884 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8449344 T + 55594921645598 T^{2} + 8449344 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 42171068827884 T^{2} + \)\(19\!\cdots\!22\)\( T^{4} + 42171068827884 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 125209641009956 T^{2} + \)\(78\!\cdots\!22\)\( T^{4} + 125209641009956 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 18246004 T + 241359719968806 T^{2} + 18246004 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−8.176804960700087156001750673924, −7.52746026032027695967384282822, −7.38897798401130006308341552390, −7.18871905068514839364563740478, −7.12210586512274957371632939359, −6.53580384491488422215690756300, −6.42302130216767427645239305831, −6.15116603126439566135620274415, −6.06556824906779349145964674417, −5.55121218188553383162781501693, −5.33744115820602921169840919180, −5.16340727838930836059552615389, −4.97208592264084290486570141198, −4.03165714156314589394822479088, −3.98932921317327353526885517996, −3.90149733788941743718702981773, −3.82574341543921192698948856484, −3.19372785401240981995850190577, −2.84828088185722445338837904774, −2.80851202304802577507559877586, −2.39203033339252285557403659578, −1.76715091672737984031257778312, −1.39743072023695782493108300958, −1.38767532699371223652205802179, −1.13582202255186179223433697523, 0, 0, 0, 0,
1.13582202255186179223433697523, 1.38767532699371223652205802179, 1.39743072023695782493108300958, 1.76715091672737984031257778312, 2.39203033339252285557403659578, 2.80851202304802577507559877586, 2.84828088185722445338837904774, 3.19372785401240981995850190577, 3.82574341543921192698948856484, 3.90149733788941743718702981773, 3.98932921317327353526885517996, 4.03165714156314589394822479088, 4.97208592264084290486570141198, 5.16340727838930836059552615389, 5.33744115820602921169840919180, 5.55121218188553383162781501693, 6.06556824906779349145964674417, 6.15116603126439566135620274415, 6.42302130216767427645239305831, 6.53580384491488422215690756300, 7.12210586512274957371632939359, 7.18871905068514839364563740478, 7.38897798401130006308341552390, 7.52746026032027695967384282822, 8.176804960700087156001750673924
