L(s) = 1 | − 2·2-s + 3·4-s + 4·7-s − 4·8-s + 9-s − 8·11-s − 8·14-s + 5·16-s − 2·18-s + 16·22-s − 6·25-s + 12·28-s + 12·29-s − 6·32-s + 3·36-s − 4·37-s + 8·43-s − 24·44-s + 9·49-s + 12·50-s − 20·53-s − 16·56-s − 24·58-s + 4·63-s + 7·64-s − 32·67-s − 16·71-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s + 1/3·9-s − 2.41·11-s − 2.13·14-s + 5/4·16-s − 0.471·18-s + 3.41·22-s − 6/5·25-s + 2.26·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.657·37-s + 1.21·43-s − 3.61·44-s + 9/7·49-s + 1.69·50-s − 2.74·53-s − 2.13·56-s − 3.15·58-s + 0.503·63-s + 7/8·64-s − 3.90·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.724491555695013698121297530874, −7.959290492018666343008631063702, −7.70223650354535578507572334847, −7.69854623831519517158383165327, −7.03828115713215538581036348773, −6.10664918750757402239876678098, −6.00899398361409811106287665786, −5.02692913607076442515975122964, −4.91686329817785172521042266230, −4.24383044940361627426015322662, −3.09122967370555540768759817180, −2.67335351129804876593036678201, −1.96145678743744747791090034638, −1.29963350151887452691131637880, 0,
1.29963350151887452691131637880, 1.96145678743744747791090034638, 2.67335351129804876593036678201, 3.09122967370555540768759817180, 4.24383044940361627426015322662, 4.91686329817785172521042266230, 5.02692913607076442515975122964, 6.00899398361409811106287665786, 6.10664918750757402239876678098, 7.03828115713215538581036348773, 7.69854623831519517158383165327, 7.70223650354535578507572334847, 7.959290492018666343008631063702, 8.724491555695013698121297530874