| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 4·13-s − 2·17-s + 2·19-s + 8·23-s + 3·25-s − 4·29-s + 2·31-s − 4·35-s − 18·37-s − 12·41-s + 2·43-s + 14·47-s − 8·49-s − 6·53-s − 8·55-s − 8·59-s − 16·61-s − 8·65-s − 8·67-s − 6·73-s + 8·77-s − 24·79-s − 2·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.458·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s + 0.359·31-s − 0.676·35-s − 2.95·37-s − 1.87·41-s + 0.304·43-s + 2.04·47-s − 8/7·49-s − 0.824·53-s − 1.07·55-s − 1.04·59-s − 2.04·61-s − 0.992·65-s − 0.977·67-s − 0.702·73-s + 0.911·77-s − 2.70·79-s − 0.219·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18 T + 152 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 60 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 107 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 152 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 192 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.513108172417896954709035390758, −8.188906288048211400673109047263, −7.52405697588398855472013158732, −7.30832317021019038057996665894, −6.96455277041837383973355613189, −6.77114367063448485009188816853, −6.11827041655018963900527463443, −5.82095200012817641448583021734, −5.27456147751351004474658486013, −5.20730402289068942368901583095, −4.65483290711681360436102256242, −4.40751171495545538595545110194, −3.38209340879253093115030979317, −3.32574850624128237928818865369, −2.66985221693776563062700877436, −2.57417546370900963034868945143, −1.58463942492643247529894348132, −1.50161031792741600254103871777, 0, 0,
1.50161031792741600254103871777, 1.58463942492643247529894348132, 2.57417546370900963034868945143, 2.66985221693776563062700877436, 3.32574850624128237928818865369, 3.38209340879253093115030979317, 4.40751171495545538595545110194, 4.65483290711681360436102256242, 5.20730402289068942368901583095, 5.27456147751351004474658486013, 5.82095200012817641448583021734, 6.11827041655018963900527463443, 6.77114367063448485009188816853, 6.96455277041837383973355613189, 7.30832317021019038057996665894, 7.52405697588398855472013158732, 8.188906288048211400673109047263, 8.513108172417896954709035390758
