| L(s) = 1 | − 2-s − 5-s + 7-s + 8-s + 10-s + 11-s − 14-s − 16-s − 4·17-s + 2·19-s − 22-s + 23-s + 25-s − 31-s + 4·34-s − 35-s − 2·37-s − 2·38-s − 40-s − 41-s − 46-s − 50-s − 55-s + 56-s + 62-s + 64-s + 70-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 7-s + 8-s + 10-s + 11-s − 14-s − 16-s − 4·17-s + 2·19-s − 22-s + 23-s + 25-s − 31-s + 4·34-s − 35-s − 2·37-s − 2·38-s − 40-s − 41-s − 46-s − 50-s − 55-s + 56-s + 62-s + 64-s + 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4876637583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4876637583\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.144276481330495505466488822654, −8.911380546109075490693015780756, −8.804816752549850898130288785367, −8.426778567277041829403347678834, −7.952754639453290614692009359621, −7.46314202071736085099869323404, −7.09678196668526935685427506810, −6.98831923034405337640474694243, −6.59549761355052160381798194968, −5.99987358125071482286251452212, −5.08253181733135584145480783251, −5.08224948295402109156648650376, −4.44218798714335602936550527273, −4.43616968648803763528943977591, −3.62932963067065786443865299221, −3.38144261975425496257365794392, −2.54528131044469447678325367096, −1.72010137694663409963575000526, −1.67755590148562703536005320934, −0.59074092560479883092033320852,
0.59074092560479883092033320852, 1.67755590148562703536005320934, 1.72010137694663409963575000526, 2.54528131044469447678325367096, 3.38144261975425496257365794392, 3.62932963067065786443865299221, 4.43616968648803763528943977591, 4.44218798714335602936550527273, 5.08224948295402109156648650376, 5.08253181733135584145480783251, 5.99987358125071482286251452212, 6.59549761355052160381798194968, 6.98831923034405337640474694243, 7.09678196668526935685427506810, 7.46314202071736085099869323404, 7.952754639453290614692009359621, 8.426778567277041829403347678834, 8.804816752549850898130288785367, 8.911380546109075490693015780756, 9.144276481330495505466488822654
