| L(s) = 1 | − 4·3-s + 15·4-s − 42·9-s − 60·12-s − 92·13-s + 161·16-s − 192·17-s − 204·23-s + 292·27-s − 60·29-s − 630·36-s + 368·39-s − 564·43-s − 644·48-s + 110·49-s + 768·51-s − 1.38e3·52-s − 744·53-s − 1.47e3·61-s + 1.45e3·64-s − 2.88e3·68-s + 816·69-s − 480·79-s + 971·81-s + 240·87-s − 3.06e3·92-s + 444·101-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 15/8·4-s − 1.55·9-s − 1.44·12-s − 1.96·13-s + 2.51·16-s − 2.73·17-s − 1.84·23-s + 2.08·27-s − 0.384·29-s − 2.91·36-s + 1.51·39-s − 2.00·43-s − 1.93·48-s + 0.320·49-s + 2.10·51-s − 3.68·52-s − 1.92·53-s − 3.09·61-s + 2.84·64-s − 5.13·68-s + 1.42·69-s − 0.683·79-s + 1.33·81-s + 0.295·87-s − 3.46·92-s + 0.437·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 92 T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 15482 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 85430 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 97842 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 282 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88290 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 372 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 387042 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 738 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 444710 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 p^{2} T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 197390 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1138950 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 374958 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 818530 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10402280165983684277881400076, −10.69979843784382227079044281776, −10.17344164674849628340590675193, −9.784261768879918723134591057729, −8.953271166992081260595141395467, −8.635542185649452984911359865494, −7.78536395774619173838981036392, −7.67209833103195052161482905257, −6.81695022730119814822014235657, −6.59105606147073290629119414619, −6.00860695015462316409116195456, −5.84071826011745873108714802504, −4.80164518488584366609914292499, −4.69423705750265306024940767360, −3.39235230899816875740440868770, −2.81394141845569709815353449003, −2.18889921538655509285145514921, −1.88370580233156522827756406761, 0, 0,
1.88370580233156522827756406761, 2.18889921538655509285145514921, 2.81394141845569709815353449003, 3.39235230899816875740440868770, 4.69423705750265306024940767360, 4.80164518488584366609914292499, 5.84071826011745873108714802504, 6.00860695015462316409116195456, 6.59105606147073290629119414619, 6.81695022730119814822014235657, 7.67209833103195052161482905257, 7.78536395774619173838981036392, 8.635542185649452984911359865494, 8.953271166992081260595141395467, 9.784261768879918723134591057729, 10.17344164674849628340590675193, 10.69979843784382227079044281776, 11.10402280165983684277881400076
