L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 4·8-s + 2·10-s − 5·11-s − 4·13-s + 8·16-s − 8·17-s − 10·19-s − 2·20-s + 10·22-s − 6·23-s + 8·26-s + 5·29-s + 9·31-s − 8·32-s + 16·34-s − 20·37-s + 20·38-s + 4·40-s − 7·41-s + 2·43-s − 10·44-s + 12·46-s − 2·47-s + 7·49-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 1.41·8-s + 0.632·10-s − 1.50·11-s − 1.10·13-s + 2·16-s − 1.94·17-s − 2.29·19-s − 0.447·20-s + 2.13·22-s − 1.25·23-s + 1.56·26-s + 0.928·29-s + 1.61·31-s − 1.41·32-s + 2.74·34-s − 3.28·37-s + 3.24·38-s + 0.632·40-s − 1.09·41-s + 0.304·43-s − 1.50·44-s + 1.76·46-s − 0.291·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.62294358370808802227290137830, −10.52755959451468544356437132672, −10.02796411348234986452558275639, −9.954365582767116809296842721188, −8.840443085737637672864224671104, −8.761550421534312872876249621153, −8.282841994599585323249472568018, −8.265369837306651636680986082671, −7.29090938036221534532074082949, −6.92315502455055926209526901200, −6.57561704634863464608148049758, −5.82378390722174223213057106595, −5.34129194771495489156303064358, −4.44088180038416228567280094359, −4.23024492524889618882193378555, −3.05208159602070366829036127000, −2.47450709069665290588655485953, −1.98958486613321615527244991942, 0, 0,
1.98958486613321615527244991942, 2.47450709069665290588655485953, 3.05208159602070366829036127000, 4.23024492524889618882193378555, 4.44088180038416228567280094359, 5.34129194771495489156303064358, 5.82378390722174223213057106595, 6.57561704634863464608148049758, 6.92315502455055926209526901200, 7.29090938036221534532074082949, 8.265369837306651636680986082671, 8.282841994599585323249472568018, 8.761550421534312872876249621153, 8.840443085737637672864224671104, 9.954365582767116809296842721188, 10.02796411348234986452558275639, 10.52755959451468544356437132672, 10.62294358370808802227290137830