| L(s) = 1 | − 4·5-s + 9-s − 12·13-s − 4·17-s + 2·25-s − 4·29-s − 20·37-s − 4·41-s − 4·45-s + 49-s − 20·53-s + 20·61-s + 48·65-s + 4·73-s + 81-s + 16·85-s + 12·89-s + 4·97-s − 20·101-s − 4·109-s − 28·113-s − 12·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s − 3.32·13-s − 0.970·17-s + 2/5·25-s − 0.742·29-s − 3.28·37-s − 0.624·41-s − 0.596·45-s + 1/7·49-s − 2.74·53-s + 2.56·61-s + 5.95·65-s + 0.468·73-s + 1/9·81-s + 1.73·85-s + 1.27·89-s + 0.406·97-s − 1.99·101-s − 0.383·109-s − 2.63·113-s − 1.10·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.034900753827151172045193415181, −7.70276610727881458590413914623, −7.20214522889553119059062687435, −6.78044077445841975719051191645, −6.77473142578757723529467681346, −5.36857771747617040863256466681, −5.33934975205348273864801508199, −4.52969287471330622027217048773, −4.44304823928958265064368122586, −3.61417461907435968748262507791, −3.27099070744047974080065927216, −2.32898343675991772184414351496, −1.90612310330978239121510440566, 0, 0,
1.90612310330978239121510440566, 2.32898343675991772184414351496, 3.27099070744047974080065927216, 3.61417461907435968748262507791, 4.44304823928958265064368122586, 4.52969287471330622027217048773, 5.33934975205348273864801508199, 5.36857771747617040863256466681, 6.77473142578757723529467681346, 6.78044077445841975719051191645, 7.20214522889553119059062687435, 7.70276610727881458590413914623, 8.034900753827151172045193415181
