L(s) = 1 | + 3-s + 9-s − 2·17-s + 25-s + 27-s − 18·41-s − 6·43-s + 11·49-s − 2·51-s − 12·59-s + 21·67-s + 3·73-s + 75-s + 81-s − 6·83-s − 26·97-s − 12·107-s + 20·113-s − 18·121-s − 18·123-s + 127-s − 6·129-s + 131-s + 137-s + 139-s + 11·147-s + 149-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.485·17-s + 1/5·25-s + 0.192·27-s − 2.81·41-s − 0.914·43-s + 11/7·49-s − 0.280·51-s − 1.56·59-s + 2.56·67-s + 0.351·73-s + 0.115·75-s + 1/9·81-s − 0.658·83-s − 2.63·97-s − 1.16·107-s + 1.88·113-s − 1.63·121-s − 1.62·123-s + 0.0887·127-s − 0.528·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.907·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−7.81954433787047912420130843770, −7.14276203698108987421908744348, −6.87083409016250175547768290582, −6.59223496572461597345140876091, −5.99827839764525534934648476274, −5.35719608952719231907701493400, −5.10333982085315791055604865036, −4.54099316603993926253598227078, −3.98286943701874096398638222736, −3.56295147327172625200245118469, −3.03473364432633697782543300813, −2.45273194171846368113264877012, −1.86758477823898263645962333276, −1.21131495596851593801387590776, 0,
1.21131495596851593801387590776, 1.86758477823898263645962333276, 2.45273194171846368113264877012, 3.03473364432633697782543300813, 3.56295147327172625200245118469, 3.98286943701874096398638222736, 4.54099316603993926253598227078, 5.10333982085315791055604865036, 5.35719608952719231907701493400, 5.99827839764525534934648476274, 6.59223496572461597345140876091, 6.87083409016250175547768290582, 7.14276203698108987421908744348, 7.81954433787047912420130843770