| L(s) = 1 | − 8.38e4·11-s + 1.67e5·13-s − 1.18e6·23-s − 2.39e6·25-s − 2.02e7·37-s − 5.24e7·47-s − 8.09e7·49-s − 3.84e7·59-s + 1.72e8·61-s + 1.96e8·71-s − 1.35e8·73-s − 6.69e8·83-s − 4.97e6·97-s − 2.58e9·107-s + 7.95e8·109-s + 3.21e8·121-s + ⋯ |
| L(s) = 1 | − 1.72·11-s + 1.62·13-s − 0.880·23-s − 1.22·25-s − 1.77·37-s − 1.56·47-s − 2.00·49-s − 0.413·59-s + 1.59·61-s + 0.919·71-s − 0.558·73-s − 1.54·83-s − 0.00570·97-s − 1.90·107-s + 0.539·109-s + 0.136·121-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 | $C_2^2 \wr C_2$ | \( 1 + 2397684 T^{2} + 223721337158 p^{2} T^{4} + 2397684 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 80990876 T^{2} + 99931934916102 p^{2} T^{4} + 80990876 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 41920 T + 2474982998 T^{2} + 41920 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 83828 T + 12244846206 T^{2} - 83828 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 392959603652 T^{2} + \)\(66\!\cdots\!98\)\( T^{4} + 392959603652 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 3354406892 p^{2} T^{2} + \)\(57\!\cdots\!82\)\( T^{4} + 3354406892 p^{20} T^{6} + p^{36} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 590976 T + 3164292620206 T^{2} + 590976 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1789692282540 T^{2} + \)\(39\!\cdots\!26\)\( T^{4} - 1789692282540 p^{18} T^{6} + p^{36} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 4016101648892 T^{2} - \)\(86\!\cdots\!46\)\( T^{4} + 4016101648892 p^{18} T^{6} + p^{36} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 10132996 T + 128627658535182 T^{2} + 10132996 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 1057904864078628 T^{2} + \)\(48\!\cdots\!82\)\( T^{4} + 1057904864078628 p^{18} T^{6} + p^{36} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 291661156978900 T^{2} + \)\(51\!\cdots\!94\)\( T^{4} - 291661156978900 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 26236544 T + 1502490430475102 T^{2} + 26236544 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 9611385835975860 T^{2} + \)\(41\!\cdots\!82\)\( T^{4} + 9611385835975860 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 19246208 T + 17337449003805110 T^{2} + 19246208 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 86181996 T + 14112924534863390 T^{2} - 86181996 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 48595610986509452 T^{2} + \)\(19\!\cdots\!78\)\( T^{4} + 48595610986509452 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 98392576 T + 59226890744783630 T^{2} - 98392576 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 67732212 T + 112599513066075446 T^{2} + 67732212 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 250248650531598524 T^{2} + \)\(43\!\cdots\!22\)\( T^{4} + 250248650531598524 p^{18} T^{6} + p^{36} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 334924608 T + 364933621530238822 T^{2} + 334924608 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 610627609923548772 T^{2} + \)\(29\!\cdots\!34\)\( T^{4} + 610627609923548772 p^{18} T^{6} + p^{36} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2487076 T + 1518499759767387654 T^{2} + 2487076 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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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
−7.79246752711959689925820647497, −7.07297444231174710839851798354, −6.93688680390265711072344548067, −6.71159213044184920291733897219, −6.65002354294228049624778450023, −6.00462098518767570661222781677, −5.88467700779542749133967414320, −5.73121175412596553525994134195, −5.60354134934156515126566768057, −5.13300591683856970138692834574, −4.79572606208111318427137474053, −4.62396599350025127171247792259, −4.59012944944254640972535520926, −3.82557784967102944329888861087, −3.58530495862992481381253870542, −3.52420824008504373014496457191, −3.52316606749792045596193453784, −2.84590888234968690151559770622, −2.51355053376568982371285627073, −2.38978843096871286307679926026, −2.03166374810893924898174771410, −1.70918195596791924817448053798, −1.26715620122702395112468811773, −1.25559596492856674127875013140, −0.919430184046490324193850763620, 0, 0, 0, 0,
0.919430184046490324193850763620, 1.25559596492856674127875013140, 1.26715620122702395112468811773, 1.70918195596791924817448053798, 2.03166374810893924898174771410, 2.38978843096871286307679926026, 2.51355053376568982371285627073, 2.84590888234968690151559770622, 3.52316606749792045596193453784, 3.52420824008504373014496457191, 3.58530495862992481381253870542, 3.82557784967102944329888861087, 4.59012944944254640972535520926, 4.62396599350025127171247792259, 4.79572606208111318427137474053, 5.13300591683856970138692834574, 5.60354134934156515126566768057, 5.73121175412596553525994134195, 5.88467700779542749133967414320, 6.00462098518767570661222781677, 6.65002354294228049624778450023, 6.71159213044184920291733897219, 6.93688680390265711072344548067, 7.07297444231174710839851798354, 7.79246752711959689925820647497
