| L(s) = 1 | + 3-s + 9-s + 8·11-s − 12·13-s + 8·23-s − 10·25-s + 27-s + 8·33-s − 4·37-s − 12·39-s − 24·47-s − 10·49-s + 8·59-s − 4·61-s + 8·69-s − 8·71-s − 20·73-s − 10·75-s + 81-s − 24·83-s − 12·97-s + 8·99-s + 24·107-s + 4·109-s − 4·111-s − 12·117-s + 26·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 2.41·11-s − 3.32·13-s + 1.66·23-s − 2·25-s + 0.192·27-s + 1.39·33-s − 0.657·37-s − 1.92·39-s − 3.50·47-s − 1.42·49-s + 1.04·59-s − 0.512·61-s + 0.963·69-s − 0.949·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 2.63·83-s − 1.21·97-s + 0.804·99-s + 2.32·107-s + 0.383·109-s − 0.379·111-s − 1.10·117-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.503024042291174545390595385787, −7.67554775307177371631314255080, −7.56237844869816331895572212869, −6.86617713649335228477722027286, −6.80910210319960057293306406499, −6.19174829158795733764599049329, −5.36858848597467135489376840463, −4.99207876302239290655432150674, −4.36074490911347519116057298154, −4.14300745369301184814326433789, −3.06807634409715093203408940942, −3.04605515966584879794132221618, −1.89290593869789230240007927288, −1.59108885184574423909298130831, 0,
1.59108885184574423909298130831, 1.89290593869789230240007927288, 3.04605515966584879794132221618, 3.06807634409715093203408940942, 4.14300745369301184814326433789, 4.36074490911347519116057298154, 4.99207876302239290655432150674, 5.36858848597467135489376840463, 6.19174829158795733764599049329, 6.80910210319960057293306406499, 6.86617713649335228477722027286, 7.56237844869816331895572212869, 7.67554775307177371631314255080, 8.503024042291174545390595385787
