| L(s) = 1 | − 8·19-s + 12·29-s + 8·31-s − 12·41-s + 10·49-s − 24·59-s + 4·61-s − 24·71-s + 16·79-s − 12·89-s − 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.83·19-s + 2.22·29-s + 1.43·31-s − 1.87·41-s + 10/7·49-s − 3.12·59-s + 0.512·61-s − 2.84·71-s + 1.80·79-s − 1.27·89-s − 1.19·101-s − 0.383·109-s − 2·121-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.379153960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379153960\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.884850748669084902261967434104, −8.414165381697922182188769224247, −8.079069396905827445691607462935, −7.65729836594056680966072047181, −7.31984579911223066795033079662, −6.65206196881078302073826784167, −6.51995456507046611306335274753, −6.31132885532463616513311594652, −5.85764403975763330722991810024, −5.23223852850501315038951118413, −4.94029586276568208421997033709, −4.37530415596092419385072835092, −4.34343037998614597722744289521, −3.74435783615640653173457378257, −3.11805689524617042229945770576, −2.70211960657649289090758403259, −2.45440663784945304908058781459, −1.61575738787133610530147912047, −1.28209248960265277118192318987, −0.35611247292156653907058253133,
0.35611247292156653907058253133, 1.28209248960265277118192318987, 1.61575738787133610530147912047, 2.45440663784945304908058781459, 2.70211960657649289090758403259, 3.11805689524617042229945770576, 3.74435783615640653173457378257, 4.34343037998614597722744289521, 4.37530415596092419385072835092, 4.94029586276568208421997033709, 5.23223852850501315038951118413, 5.85764403975763330722991810024, 6.31132885532463616513311594652, 6.51995456507046611306335274753, 6.65206196881078302073826784167, 7.31984579911223066795033079662, 7.65729836594056680966072047181, 8.079069396905827445691607462935, 8.414165381697922182188769224247, 8.884850748669084902261967434104
