| L(s) = 1 | − 2·5-s + 4·7-s + 4·11-s − 2·13-s − 4·17-s − 4·19-s − 8·23-s + 3·25-s − 8·29-s − 4·31-s − 8·35-s − 12·41-s + 8·43-s + 4·47-s − 2·49-s − 12·53-s − 8·55-s − 4·59-s + 8·61-s + 4·65-s − 20·67-s + 12·71-s + 16·77-s − 4·83-s + 8·85-s − 12·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.20·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.66·23-s + 3/5·25-s − 1.48·29-s − 0.718·31-s − 1.35·35-s − 1.87·41-s + 1.21·43-s + 0.583·47-s − 2/7·49-s − 1.64·53-s − 1.07·55-s − 0.520·59-s + 1.02·61-s + 0.496·65-s − 2.44·67-s + 1.42·71-s + 1.82·77-s − 0.439·83-s + 0.867·85-s − 1.27·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 172 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.168981303032572577980621066349, −7.84751013594059444218594388666, −7.33469462211241109564909565564, −7.26430439712068805328148568910, −6.60484018804872206566772258666, −6.47573440382732111789257703802, −5.87746010886331827597204936922, −5.62407627780928431427930542830, −4.98700068507847448250284164090, −4.72854018272802603278693727786, −4.26837272076479963526390853975, −4.18724864546697459658183215267, −3.51915981516653765988320901798, −3.43363495718893374460214150538, −2.35164778282133133987942816859, −2.24704576644844845324494842483, −1.49108367490569580390339329579, −1.43019927326815246478987455324, 0, 0,
1.43019927326815246478987455324, 1.49108367490569580390339329579, 2.24704576644844845324494842483, 2.35164778282133133987942816859, 3.43363495718893374460214150538, 3.51915981516653765988320901798, 4.18724864546697459658183215267, 4.26837272076479963526390853975, 4.72854018272802603278693727786, 4.98700068507847448250284164090, 5.62407627780928431427930542830, 5.87746010886331827597204936922, 6.47573440382732111789257703802, 6.60484018804872206566772258666, 7.26430439712068805328148568910, 7.33469462211241109564909565564, 7.84751013594059444218594388666, 8.168981303032572577980621066349
