| L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 8-s + 2·10-s + 14-s + 3·16-s − 3·17-s + 7·19-s − 2·20-s + 6·23-s − 2·25-s − 28-s + 29-s + 3·32-s − 3·34-s + 2·35-s + 7·38-s − 2·40-s + 6·46-s − 6·49-s − 2·50-s − 56-s + 58-s + 7·61-s − 5·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s + 0.267·14-s + 3/4·16-s − 0.727·17-s + 1.60·19-s − 0.447·20-s + 1.25·23-s − 2/5·25-s − 0.188·28-s + 0.185·29-s + 0.530·32-s − 0.514·34-s + 0.338·35-s + 1.13·38-s − 0.316·40-s + 0.884·46-s − 6/7·49-s − 0.282·50-s − 0.133·56-s + 0.131·58-s + 0.896·61-s − 5/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.849662104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.849662104\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 23 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.995253530904751133518055375810, −8.506468272495093857903463724004, −7.942708184620208177586607803173, −7.54518963929364082402296439919, −6.92677160844862320947367634752, −6.46696578825741591859713594869, −5.84172766657423632102261961472, −5.39225266183481394442134963720, −5.01223535987614904808614208315, −4.63219413339306109843413540997, −3.88416873941547286398962122207, −3.34988977695058998804657957077, −2.68882789652280471010753690810, −1.84951842224862942364321276305, −0.994782997549843381002508282820,
0.994782997549843381002508282820, 1.84951842224862942364321276305, 2.68882789652280471010753690810, 3.34988977695058998804657957077, 3.88416873941547286398962122207, 4.63219413339306109843413540997, 5.01223535987614904808614208315, 5.39225266183481394442134963720, 5.84172766657423632102261961472, 6.46696578825741591859713594869, 6.92677160844862320947367634752, 7.54518963929364082402296439919, 7.942708184620208177586607803173, 8.506468272495093857903463724004, 8.995253530904751133518055375810
