| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s + 4·10-s − 10·13-s − 4·16-s + 16·17-s + 4·20-s + 3·25-s − 20·26-s − 4·29-s − 8·32-s + 32·34-s + 10·37-s + 20·41-s − 5·49-s + 6·50-s − 20·52-s + 4·53-s − 8·58-s + 14·61-s − 8·64-s − 20·65-s + 32·68-s − 10·73-s + 20·74-s − 8·80-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s + 1.26·10-s − 2.77·13-s − 16-s + 3.88·17-s + 0.894·20-s + 3/5·25-s − 3.92·26-s − 0.742·29-s − 1.41·32-s + 5.48·34-s + 1.64·37-s + 3.12·41-s − 5/7·49-s + 0.848·50-s − 2.77·52-s + 0.549·53-s − 1.05·58-s + 1.79·61-s − 64-s − 2.48·65-s + 3.88·68-s − 1.17·73-s + 2.32·74-s − 0.894·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.293649675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.293649675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.209731645846489900441455328004, −8.033038279448884875142132147382, −7.72379481902030800570658235358, −7.49513099271809325678009295871, −6.88066201922308126532422526866, −6.25177698744351379012073167484, −5.60485752225476628583929294992, −5.43765056456230093289829322534, −5.21952785456145416033046309322, −4.39646975608954166240974959735, −3.97301637909484367451284003600, −3.07119386896251421193504749200, −2.76209099156017414000056665621, −2.20929616241085403285528735477, −1.02045420469944533258202986585,
1.02045420469944533258202986585, 2.20929616241085403285528735477, 2.76209099156017414000056665621, 3.07119386896251421193504749200, 3.97301637909484367451284003600, 4.39646975608954166240974959735, 5.21952785456145416033046309322, 5.43765056456230093289829322534, 5.60485752225476628583929294992, 6.25177698744351379012073167484, 6.88066201922308126532422526866, 7.49513099271809325678009295871, 7.72379481902030800570658235358, 8.033038279448884875142132147382, 9.209731645846489900441455328004
