| L(s) = 1 | + 3-s − 5-s − 7-s − 2·13-s − 15-s − 17-s − 4·19-s − 21-s − 23-s + 5·25-s − 27-s − 8·29-s + 8·31-s + 35-s − 10·37-s − 2·39-s − 4·41-s − 3·47-s − 6·49-s − 51-s + 9·53-s − 4·57-s − 12·59-s + 2·65-s + 11·67-s − 69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 0.554·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.320·39-s − 0.624·41-s − 0.437·47-s − 6/7·49-s − 0.140·51-s + 1.23·53-s − 0.529·57-s − 1.56·59-s + 0.248·65-s + 1.34·67-s − 0.120·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9773321464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9773321464\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.636933176785451904069438070314, −8.808034749425798188478212208309, −8.682564242250173785280495127017, −8.268410522261631552311170669465, −7.937494018584354332412640439396, −7.37831638693136443353044703961, −7.13720998966978627417700528126, −6.53567046125858642243839030671, −6.50764500493989193886828616054, −5.77221846138166239816214610257, −5.33579426698081216509373673756, −4.87561858807177483734283231305, −4.47755303108085013141610458916, −3.88794538460450066456845519236, −3.63512667413612941284936930960, −2.93111761575036409298880115791, −2.70445935211182784421223257139, −1.97271493124037895640001122717, −1.47151040409163637911748043097, −0.33648798337370694726029718505,
0.33648798337370694726029718505, 1.47151040409163637911748043097, 1.97271493124037895640001122717, 2.70445935211182784421223257139, 2.93111761575036409298880115791, 3.63512667413612941284936930960, 3.88794538460450066456845519236, 4.47755303108085013141610458916, 4.87561858807177483734283231305, 5.33579426698081216509373673756, 5.77221846138166239816214610257, 6.50764500493989193886828616054, 6.53567046125858642243839030671, 7.13720998966978627417700528126, 7.37831638693136443353044703961, 7.937494018584354332412640439396, 8.268410522261631552311170669465, 8.682564242250173785280495127017, 8.808034749425798188478212208309, 9.636933176785451904069438070314
