| L(s) = 1 | − 3.48e4·9-s − 3.48e4·11-s + 2.27e5·19-s − 6.26e6·29-s + 3.74e5·31-s − 1.76e7·41-s − 1.53e8·49-s − 4.38e8·59-s − 1.03e8·61-s − 5.04e8·71-s − 1.79e9·79-s + 3.16e8·81-s − 2.38e9·89-s + 1.21e9·99-s − 1.46e9·101-s − 2.38e9·109-s − 3.67e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.40e10·169-s + ⋯ |
| L(s) = 1 | − 1.77·9-s − 0.716·11-s + 0.400·19-s − 1.64·29-s + 0.0728·31-s − 0.975·41-s − 3.79·49-s − 4.71·59-s − 0.952·61-s − 2.35·71-s − 5.18·79-s + 0.817·81-s − 4.02·89-s + 1.26·99-s − 1.39·101-s − 1.61·109-s − 1.55·121-s − 1.32·169-s − 0.709·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 34844 T^{2} + 99693718 p^{2} T^{4} + 34844 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 153156620 T^{2} + 186133228268502 p^{2} T^{4} + 153156620 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 17400 T + 2292818982 T^{2} + 17400 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 14000436724 T^{2} + 83025702170976147702 T^{4} + 14000436724 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 181978394436 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} + 181978394436 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 113832 T + 402567657014 T^{2} - 113832 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 820282526580 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} - 820282526580 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 3132828 T + 12854589500734 T^{2} + 3132828 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 187232 T + 40099942836798 T^{2} - 187232 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 462603258659540 T^{2} + \)\(87\!\cdots\!58\)\( T^{4} + 462603258659540 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8824068 T + 671552976228678 T^{2} + 8824068 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 358707990293372 T^{2} + \)\(21\!\cdots\!94\)\( T^{4} + 358707990293372 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 1037666802860076 T^{2} + \)\(16\!\cdots\!22\)\( T^{4} + 1037666802860076 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 5310844341928340 T^{2} + \)\(28\!\cdots\!78\)\( T^{4} + 5310844341928340 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 219497736 T + 28852098295168902 T^{2} + 219497736 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 51522236 T - 2665246277981394 T^{2} + 51522236 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 83417417389239260 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} + 83417417389239260 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 252040944 T + 101042798798695246 T^{2} + 252040944 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 79546297979572004 T^{2} + \)\(52\!\cdots\!42\)\( T^{4} + 79546297979572004 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 897477504 T + 421888354983619742 T^{2} + 897477504 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 186261173343564060 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + 186261173343564060 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1190659092 T + 825013495390166934 T^{2} + 1190659092 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 899946211039106180 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} + 899946211039106180 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−9.248690874930344870637493131437, −8.409704462520833552283421999892, −8.264079406759189964191743693802, −8.225316738347379944204700366381, −8.010412149905111408038730621593, −7.41076614735694529055843775186, −7.23520229078054879739179986574, −6.93552088905223130002956038814, −6.62827015041767933920774676348, −6.04245048185046693151732076587, −5.80791848513887434309983924910, −5.76650912898009859723414577984, −5.47076809906182606112948496482, −5.04464868789496163379738809733, −4.52856792989392114875678591492, −4.37677346063179779758774703002, −4.17796730730224418649240176930, −3.17432531012975217295712432134, −3.10626410255249411322493198718, −2.99264735407795847804303093243, −2.94898253623659276550945996913, −1.99205205295144250312812237282, −1.78713058895031423833281549178, −1.37340889369012243545723946132, −1.22541580545675766832712117431, 0, 0, 0, 0,
1.22541580545675766832712117431, 1.37340889369012243545723946132, 1.78713058895031423833281549178, 1.99205205295144250312812237282, 2.94898253623659276550945996913, 2.99264735407795847804303093243, 3.10626410255249411322493198718, 3.17432531012975217295712432134, 4.17796730730224418649240176930, 4.37677346063179779758774703002, 4.52856792989392114875678591492, 5.04464868789496163379738809733, 5.47076809906182606112948496482, 5.76650912898009859723414577984, 5.80791848513887434309983924910, 6.04245048185046693151732076587, 6.62827015041767933920774676348, 6.93552088905223130002956038814, 7.23520229078054879739179986574, 7.41076614735694529055843775186, 8.010412149905111408038730621593, 8.225316738347379944204700366381, 8.264079406759189964191743693802, 8.409704462520833552283421999892, 9.248690874930344870637493131437
