| L(s) = 1 | + 3-s − 2·4-s + 9-s − 2·12-s + 4·16-s − 2·19-s − 8·25-s + 27-s − 2·36-s + 8·43-s + 4·48-s + 6·49-s − 2·57-s − 8·64-s − 16·67-s − 24·73-s − 8·75-s + 4·76-s + 81-s + 12·97-s + 16·100-s − 2·108-s − 6·121-s + 127-s + 8·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 0.577·12-s + 16-s − 0.458·19-s − 8/5·25-s + 0.192·27-s − 1/3·36-s + 1.21·43-s + 0.577·48-s + 6/7·49-s − 0.264·57-s − 64-s − 1.95·67-s − 2.80·73-s − 0.923·75-s + 0.458·76-s + 1/9·81-s + 1.21·97-s + 8/5·100-s − 0.192·108-s − 0.545·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.146218163409257504246896511696, −7.75874489116368219190333094072, −7.46948111262134750676001885536, −6.95342660923974710678587001035, −6.17193319541139457653871321098, −5.86419362513548861116095541244, −5.45724855145797667979830685448, −4.68218539424631870153596716604, −4.33386863833412742892715424471, −3.93246989003627989923612947570, −3.34783349995921635939576352452, −2.72447220511300502563204735741, −2.00669945433898559128893684071, −1.20470685171656408945477638185, 0,
1.20470685171656408945477638185, 2.00669945433898559128893684071, 2.72447220511300502563204735741, 3.34783349995921635939576352452, 3.93246989003627989923612947570, 4.33386863833412742892715424471, 4.68218539424631870153596716604, 5.45724855145797667979830685448, 5.86419362513548861116095541244, 6.17193319541139457653871321098, 6.95342660923974710678587001035, 7.46948111262134750676001885536, 7.75874489116368219190333094072, 8.146218163409257504246896511696
