L(s) = 1 | − 3-s − 4·7-s + 9-s − 12·13-s + 4·21-s − 10·25-s − 27-s − 20·31-s − 4·37-s + 12·39-s + 16·43-s − 2·49-s − 4·61-s − 4·63-s + 8·67-s − 20·73-s + 10·75-s + 12·79-s + 81-s + 48·91-s + 20·93-s − 12·97-s + 20·103-s + 4·109-s + 4·111-s − 12·117-s − 6·121-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 3.32·13-s + 0.872·21-s − 2·25-s − 0.192·27-s − 3.59·31-s − 0.657·37-s + 1.92·39-s + 2.43·43-s − 2/7·49-s − 0.512·61-s − 0.503·63-s + 0.977·67-s − 2.34·73-s + 1.15·75-s + 1.35·79-s + 1/9·81-s + 5.03·91-s + 2.07·93-s − 1.21·97-s + 1.97·103-s + 0.383·109-s + 0.379·111-s − 1.10·117-s − 0.545·121-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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−7.73185283482156320096864018539, −7.56237844869816331895572212869, −7.29744951786402483785793531451, −6.86617713649335228477722027286, −6.19174829158795733764599049329, −5.81271121311664380914911812156, −5.25073137830748218691985779847, −4.99207876302239290655432150674, −4.14300745369301184814326433789, −3.77490371552045331982722842815, −3.04605515966584879794132221618, −2.36547685273488160999702672858, −1.89290593869789230240007927288, 0, 0,
1.89290593869789230240007927288, 2.36547685273488160999702672858, 3.04605515966584879794132221618, 3.77490371552045331982722842815, 4.14300745369301184814326433789, 4.99207876302239290655432150674, 5.25073137830748218691985779847, 5.81271121311664380914911812156, 6.19174829158795733764599049329, 6.86617713649335228477722027286, 7.29744951786402483785793531451, 7.56237844869816331895572212869, 7.73185283482156320096864018539