| L(s) = 1 | − 2·3-s − 8·5-s + 9-s + 16·15-s + 4·19-s − 16·23-s + 38·25-s + 4·27-s + 4·29-s − 16·43-s − 8·45-s + 8·47-s + 49-s − 20·53-s − 8·57-s + 24·67-s + 32·69-s − 28·73-s − 76·75-s − 11·81-s − 8·87-s − 32·95-s − 4·97-s + 24·101-s + 128·115-s − 22·121-s − 136·125-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3.57·5-s + 1/3·9-s + 4.13·15-s + 0.917·19-s − 3.33·23-s + 38/5·25-s + 0.769·27-s + 0.742·29-s − 2.43·43-s − 1.19·45-s + 1.16·47-s + 1/7·49-s − 2.74·53-s − 1.05·57-s + 2.93·67-s + 3.85·69-s − 3.27·73-s − 8.77·75-s − 1.22·81-s − 0.857·87-s − 3.28·95-s − 0.406·97-s + 2.38·101-s + 11.9·115-s − 2·121-s − 12.1·125-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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.092660011067177558429540865439, −7.54801754874801697338809881828, −7.48523560915642098619060234168, −6.77420851319528696059978803031, −6.34948564084072494191115955004, −5.85675979257035438626438847546, −4.88658861407619853654072156509, −4.85762247801381754805120738895, −4.20793108421579356205513357787, −3.61118272831897977078492843408, −3.56270542712905614063510381286, −2.65043598386331574261041648988, −1.21717442621058177691884331097, 0, 0,
1.21717442621058177691884331097, 2.65043598386331574261041648988, 3.56270542712905614063510381286, 3.61118272831897977078492843408, 4.20793108421579356205513357787, 4.85762247801381754805120738895, 4.88658861407619853654072156509, 5.85675979257035438626438847546, 6.34948564084072494191115955004, 6.77420851319528696059978803031, 7.48523560915642098619060234168, 7.54801754874801697338809881828, 8.092660011067177558429540865439
