| L(s) = 1 | + 4·11-s + 2·13-s − 4·17-s − 8·23-s − 2·25-s + 4·29-s + 8·31-s + 4·37-s − 16·41-s + 8·43-s − 12·47-s − 6·49-s − 4·53-s − 4·59-s − 4·61-s + 8·67-s + 4·71-s + 12·73-s + 4·83-s − 24·89-s − 4·97-s + 4·101-s − 16·103-s + 12·109-s − 12·113-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.554·13-s − 0.970·17-s − 1.66·23-s − 2/5·25-s + 0.742·29-s + 1.43·31-s + 0.657·37-s − 2.49·41-s + 1.21·43-s − 1.75·47-s − 6/7·49-s − 0.549·53-s − 0.520·59-s − 0.512·61-s + 0.977·67-s + 0.474·71-s + 1.40·73-s + 0.439·83-s − 2.54·89-s − 0.406·97-s + 0.398·101-s − 1.57·103-s + 1.14·109-s − 1.12·113-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782855521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782855521\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.150911798426332407790101882222, −8.012236590384725700919024600428, −7.10139367489071383235333362411, −7.09726733062774080796858753725, −6.46204105867004590275406219758, −6.41886824002976433265160466270, −6.20713227332736538577493049552, −5.67068367419872630669135561657, −5.26532071508072382387566821565, −4.77534861952468919138204559635, −4.41691050265947322150110429761, −4.25997856206764745082617805144, −3.62184367866548671056221486186, −3.53971322623337477735449375956, −2.88200276054860310120281992283, −2.52972675807092789999728380822, −1.80567599464988680003150126157, −1.69331090941326449769272159303, −1.06957032174891951723024482991, −0.33772794923970751892016268490,
0.33772794923970751892016268490, 1.06957032174891951723024482991, 1.69331090941326449769272159303, 1.80567599464988680003150126157, 2.52972675807092789999728380822, 2.88200276054860310120281992283, 3.53971322623337477735449375956, 3.62184367866548671056221486186, 4.25997856206764745082617805144, 4.41691050265947322150110429761, 4.77534861952468919138204559635, 5.26532071508072382387566821565, 5.67068367419872630669135561657, 6.20713227332736538577493049552, 6.41886824002976433265160466270, 6.46204105867004590275406219758, 7.09726733062774080796858753725, 7.10139367489071383235333362411, 8.012236590384725700919024600428, 8.150911798426332407790101882222
