| L(s) = 1 | + 2·2-s + 2·4-s − 5·9-s − 4·16-s + 2·17-s − 10·18-s + 9·25-s − 8·32-s + 4·34-s − 10·36-s − 12·43-s + 10·49-s + 18·50-s + 10·59-s − 8·64-s + 14·67-s + 4·68-s + 16·81-s − 12·83-s − 24·86-s − 30·89-s + 20·98-s + 18·100-s + 20·118-s + 121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 5/3·9-s − 16-s + 0.485·17-s − 2.35·18-s + 9/5·25-s − 1.41·32-s + 0.685·34-s − 5/3·36-s − 1.82·43-s + 10/7·49-s + 2.54·50-s + 1.30·59-s − 64-s + 1.71·67-s + 0.485·68-s + 16/9·81-s − 1.31·83-s − 2.58·86-s − 3.17·89-s + 2.02·98-s + 9/5·100-s + 1.84·118-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2238016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2238016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175371117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175371117\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.57009324897863038638993460964, −7.02899684736018290765739211981, −6.73206764436763783992136345255, −6.38432729046312167846494136123, −5.73983821519585618071951920597, −5.51689034756848009242917184884, −5.19615418812243182959644282972, −4.77841787015695241301942094368, −4.17254224132565568475743597363, −3.68387039714137040598591468375, −3.25318366564901690202350486989, −2.64468366884270875657211175727, −2.60165676189877959002986810388, −1.57878507569022462848881957770, −0.54718724832019521121701411078,
0.54718724832019521121701411078, 1.57878507569022462848881957770, 2.60165676189877959002986810388, 2.64468366884270875657211175727, 3.25318366564901690202350486989, 3.68387039714137040598591468375, 4.17254224132565568475743597363, 4.77841787015695241301942094368, 5.19615418812243182959644282972, 5.51689034756848009242917184884, 5.73983821519585618071951920597, 6.38432729046312167846494136123, 6.73206764436763783992136345255, 7.02899684736018290765739211981, 7.57009324897863038638993460964
