| L(s) = 1 | − 5·9-s − 2·13-s − 10·17-s − 10·25-s + 6·29-s − 4·37-s − 16·41-s − 5·49-s − 18·53-s − 28·61-s + 18·73-s + 16·81-s − 24·89-s + 28·97-s + 28·101-s − 14·109-s − 36·113-s + 10·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 50·153-s + 157-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 0.554·13-s − 2.42·17-s − 2·25-s + 1.11·29-s − 0.657·37-s − 2.49·41-s − 5/7·49-s − 2.47·53-s − 3.58·61-s + 2.10·73-s + 16/9·81-s − 2.54·89-s + 2.84·97-s + 2.78·101-s − 1.34·109-s − 3.38·113-s + 0.924·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 4.04·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.74515606034087698025852781244, −6.73715959969007181198215210739, −6.72377104864689556277323751664, −6.17043864352631516446211866530, −5.90719710332910393553929521875, −5.03277316009204668285879882866, −4.99527268882455447172353841146, −4.41313931480294198592546395456, −3.81156163044135505245852369486, −3.09698588528238924660584466755, −2.89182953598080292522833763992, −2.01374174404058579498750722555, −1.78091799751097958894751684261, 0, 0,
1.78091799751097958894751684261, 2.01374174404058579498750722555, 2.89182953598080292522833763992, 3.09698588528238924660584466755, 3.81156163044135505245852369486, 4.41313931480294198592546395456, 4.99527268882455447172353841146, 5.03277316009204668285879882866, 5.90719710332910393553929521875, 6.17043864352631516446211866530, 6.72377104864689556277323751664, 6.73715959969007181198215210739, 7.74515606034087698025852781244
