| L(s) = 1 | + 2·7-s − 25-s + 3·31-s − 37-s − 2·43-s + 3·49-s + 3·61-s − 67-s − 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·7-s − 25-s + 3·31-s − 37-s − 2·43-s + 3·49-s + 3·61-s − 67-s − 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.890045643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890045643\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.827445410240622499585937216638, −8.461873063680084174772506232162, −8.390496838673656267932891779892, −8.210396102352423138216133172160, −7.52804843762181266144291761352, −7.39520608821714860115874274368, −6.85500296758258296281903152674, −6.52180761045433012056834322882, −5.94795617781638364497098502006, −5.68610442572027824444139252728, −5.01267056771897243770696118081, −4.99154630863048266366985266537, −4.41592211344093801552206808659, −4.21365371673157062921686257434, −3.50939181976400416816343747200, −3.14130252537827244101300688766, −2.23824506610150294948879339969, −2.21163386245181703584064740596, −1.44705810549119933404593960722, −0.947690568544799367276752240675,
0.947690568544799367276752240675, 1.44705810549119933404593960722, 2.21163386245181703584064740596, 2.23824506610150294948879339969, 3.14130252537827244101300688766, 3.50939181976400416816343747200, 4.21365371673157062921686257434, 4.41592211344093801552206808659, 4.99154630863048266366985266537, 5.01267056771897243770696118081, 5.68610442572027824444139252728, 5.94795617781638364497098502006, 6.52180761045433012056834322882, 6.85500296758258296281903152674, 7.39520608821714860115874274368, 7.52804843762181266144291761352, 8.210396102352423138216133172160, 8.390496838673656267932891779892, 8.461873063680084174772506232162, 8.827445410240622499585937216638
