| L(s) = 1 | − 8.49e3·13-s + 2.17e4·25-s − 1.55e5·37-s + 3.44e5·49-s − 2.70e5·61-s + 4.93e5·73-s + 4.84e6·97-s − 2.66e6·109-s + 6.83e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.57e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.86·13-s + 1.39·25-s − 3.06·37-s + 2.92·49-s − 1.19·61-s + 1.26·73-s + 5.30·97-s − 2.05·109-s + 3.85·121-s + 5.33·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.528891428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528891428\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( ( 1 - 2174 p T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 172223 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 3416762 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2123 T + p^{6} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 25993658 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 59919913 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 240347018 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 299042638 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1134124994 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 38773 T + p^{6} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 4474196638 T^{2} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 11252361746 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 12684799658 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 44327205938 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 28144893242 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 67573 T + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 101378868551 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 127300731842 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 123337 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 398821906079 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 188526497902 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 390541984922 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1210969 T + p^{6} T^{2} )^{4} \) |
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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
−7.26448844567068459155793185631, −6.83973465640490218493611352348, −6.58143909676555471812286659999, −6.40809950544938373135686617913, −6.16134097824679207741310566073, −5.52919086805709153144128930551, −5.43678967955246442610712952542, −5.25503059414365800906744555943, −5.08422069697187097391925689601, −4.82380207655423120717155604187, −4.45883141955837928000648334158, −4.33736940119163083352039085872, −4.07119721072974313869549276486, −3.47684643252448983928749064464, −3.33198229031233733891979216149, −2.94179909881492989719700652789, −2.88566986105822835565492810597, −2.27473689767772040227547714973, −2.23731835475715490442312097510, −1.85180264098914872945070718376, −1.81394751680368106050058599062, −0.997656415426888858435432705246, −0.806198653482410573783326423562, −0.39863640025025660917817325219, −0.26554842696427016150909741192,
0.26554842696427016150909741192, 0.39863640025025660917817325219, 0.806198653482410573783326423562, 0.997656415426888858435432705246, 1.81394751680368106050058599062, 1.85180264098914872945070718376, 2.23731835475715490442312097510, 2.27473689767772040227547714973, 2.88566986105822835565492810597, 2.94179909881492989719700652789, 3.33198229031233733891979216149, 3.47684643252448983928749064464, 4.07119721072974313869549276486, 4.33736940119163083352039085872, 4.45883141955837928000648334158, 4.82380207655423120717155604187, 5.08422069697187097391925689601, 5.25503059414365800906744555943, 5.43678967955246442610712952542, 5.52919086805709153144128930551, 6.16134097824679207741310566073, 6.40809950544938373135686617913, 6.58143909676555471812286659999, 6.83973465640490218493611352348, 7.26448844567068459155793185631
