| L(s) = 1 | + 8·11-s + 16·19-s − 12·29-s + 12·41-s + 14·49-s − 24·59-s + 28·61-s − 16·79-s + 4·89-s + 28·101-s − 28·109-s + 26·121-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 + 193-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 3.67·19-s − 2.22·29-s + 1.87·41-s + 2·49-s − 3.12·59-s + 3.58·61-s − 1.80·79-s + 0.423·89-s + 2.78·101-s − 2.68·109-s + 2.36·121-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 + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.392341531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.392341531\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−7.79565758353505274496884020399, −7.66207430316753688748534729164, −7.44610161722978143922597859654, −7.19963041675976963399598464222, −6.61615527837999324447270766584, −6.51029837377020657996277402207, −5.84565615372494755067694723813, −5.72706747199245297884750902979, −5.23952919251081496661943120189, −5.16814582971103776287837552771, −4.29694640681980916209141146769, −4.14065374167586685781813738531, −3.66638445494003153698409979302, −3.58376127349263816501765651718, −2.86514723193796199652280684450, −2.71294551667721947694464722278, −1.73444234612731094140863624316, −1.61885372727862841894104865554, −0.932513889295250556769573727180, −0.72201391453829937284079479117,
0.72201391453829937284079479117, 0.932513889295250556769573727180, 1.61885372727862841894104865554, 1.73444234612731094140863624316, 2.71294551667721947694464722278, 2.86514723193796199652280684450, 3.58376127349263816501765651718, 3.66638445494003153698409979302, 4.14065374167586685781813738531, 4.29694640681980916209141146769, 5.16814582971103776287837552771, 5.23952919251081496661943120189, 5.72706747199245297884750902979, 5.84565615372494755067694723813, 6.51029837377020657996277402207, 6.61615527837999324447270766584, 7.19963041675976963399598464222, 7.44610161722978143922597859654, 7.66207430316753688748534729164, 7.79565758353505274496884020399
