| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s + 2·11-s − 4·14-s − 16-s − 2·22-s − 2·23-s − 2·25-s − 4·28-s + 2·29-s − 5·32-s + 18·37-s + 12·43-s − 2·44-s + 2·46-s + 9·49-s + 2·50-s + 6·53-s + 12·56-s − 2·58-s + 7·64-s + 26·67-s + 8·71-s − 18·74-s + 8·77-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s + 0.603·11-s − 1.06·14-s − 1/4·16-s − 0.426·22-s − 0.417·23-s − 2/5·25-s − 0.755·28-s + 0.371·29-s − 0.883·32-s + 2.95·37-s + 1.82·43-s − 0.301·44-s + 0.294·46-s + 9/7·49-s + 0.282·50-s + 0.824·53-s + 1.60·56-s − 0.262·58-s + 7/8·64-s + 3.17·67-s + 0.949·71-s − 2.09·74-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.641963702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641963702\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
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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
−8.257592101911283744560591701545, −7.890928130557456364227381514026, −7.62359593961898260762269695077, −7.16362447456422089253667230116, −6.56388949099519534552005448412, −5.91309541673598571192107349872, −5.61177704237856038733327591668, −4.94669007641676557987915729580, −4.55453264691405256189393135265, −4.05071730270296868163585158915, −3.82628263447503921016397659010, −2.63494432533946030601362920276, −2.19187755603574590872811791043, −1.31728455434506809559065669406, −0.825035702402623668906810817120,
0.825035702402623668906810817120, 1.31728455434506809559065669406, 2.19187755603574590872811791043, 2.63494432533946030601362920276, 3.82628263447503921016397659010, 4.05071730270296868163585158915, 4.55453264691405256189393135265, 4.94669007641676557987915729580, 5.61177704237856038733327591668, 5.91309541673598571192107349872, 6.56388949099519534552005448412, 7.16362447456422089253667230116, 7.62359593961898260762269695077, 7.890928130557456364227381514026, 8.257592101911283744560591701545
