| L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·19-s + 2·21-s + 10·25-s − 4·27-s + 8·31-s − 4·37-s + 49-s + 8·57-s + 63-s + 20·75-s − 11·81-s + 16·93-s + 8·103-s + 4·109-s − 8·111-s + 22·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 2·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.917·19-s + 0.436·21-s + 2·25-s − 0.769·27-s + 1.43·31-s − 0.657·37-s + 1/7·49-s + 1.05·57-s + 0.125·63-s + 2.30·75-s − 1.22·81-s + 1.65·93-s + 0.788·103-s + 0.383·109-s − 0.759·111-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + 0.0848·139-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438570245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438570245\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.324992495285741284362677030194, −7.88228085215062079869423210648, −7.43488005344851259695231195874, −7.07902687157602483250869928969, −6.54854426954325821739973185181, −6.06669220110939077121733120966, −5.43958938429240367681396042537, −4.94594526346176512640726185539, −4.59738920263792120538814303370, −3.90646639086371242340910041967, −3.33459863544371555082653540098, −2.91189318717029163234282638486, −2.45622350537639496068289705098, −1.65740511214701006390930450507, −0.884537711243478395059635587644,
0.884537711243478395059635587644, 1.65740511214701006390930450507, 2.45622350537639496068289705098, 2.91189318717029163234282638486, 3.33459863544371555082653540098, 3.90646639086371242340910041967, 4.59738920263792120538814303370, 4.94594526346176512640726185539, 5.43958938429240367681396042537, 6.06669220110939077121733120966, 6.54854426954325821739973185181, 7.07902687157602483250869928969, 7.43488005344851259695231195874, 7.88228085215062079869423210648, 8.324992495285741284362677030194
