L(s) = 1 | − 4-s − 3·16-s + 8·19-s − 5·25-s − 16·31-s − 4·61-s + 7·64-s − 8·76-s − 32·79-s + 5·100-s + 28·109-s − 22·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 3/4·16-s + 1.83·19-s − 25-s − 2.87·31-s − 0.512·61-s + 7/8·64-s − 0.917·76-s − 3.60·79-s + 1/2·100-s + 2.68·109-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.006239800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.006239800\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.459501872335897529754981238442, −9.007523399322196319546071085346, −8.600807646800967871459093888562, −7.949009327963947257185077913757, −7.77182765048689566584365085880, −7.27481775856082839766551623409, −7.03577164825596187182892335619, −6.66161758916153036436930111346, −5.81923015699735430038580931572, −5.71907810840089063330202393637, −5.39105757960779777521960309001, −4.85524717981132801762920878720, −4.34238186933368609338011079235, −4.04715576888922035705059558833, −3.29610816047882424687748144418, −3.28581702853468131659674010390, −2.41924016286910073489575241045, −1.84868146419156580856953688642, −1.34085808902525001930383209282, −0.35875281877674739545711697242,
0.35875281877674739545711697242, 1.34085808902525001930383209282, 1.84868146419156580856953688642, 2.41924016286910073489575241045, 3.28581702853468131659674010390, 3.29610816047882424687748144418, 4.04715576888922035705059558833, 4.34238186933368609338011079235, 4.85524717981132801762920878720, 5.39105757960779777521960309001, 5.71907810840089063330202393637, 5.81923015699735430038580931572, 6.66161758916153036436930111346, 7.03577164825596187182892335619, 7.27481775856082839766551623409, 7.77182765048689566584365085880, 7.949009327963947257185077913757, 8.600807646800967871459093888562, 9.007523399322196319546071085346, 9.459501872335897529754981238442