| L(s) = 1 | + 2·7-s + 10·13-s + 14·19-s − 10·25-s + 8·31-s + 22·37-s − 16·43-s − 11·49-s − 2·61-s − 10·67-s − 14·73-s − 34·79-s + 20·91-s − 38·97-s + 26·103-s + 4·109-s − 22·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2.77·13-s + 3.21·19-s − 2·25-s + 1.43·31-s + 3.61·37-s − 2.43·43-s − 1.57·49-s − 0.256·61-s − 1.22·67-s − 1.63·73-s − 3.82·79-s + 2.09·91-s − 3.85·97-s + 2.56·103-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 2.42·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340759355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340759355\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64964189773580716903063683793, −13.03049201961870670350835369127, −13.03049201961870670350835369127, −11.70175137554538908977854303538, −11.70175137554538908977854303538, −11.38842166918163472638466197964, −11.38842166918163472638466197964, −10.22208553622788501090350672882, −10.22208553622788501090350672882, −9.387437094399487619618753359356, −9.387437094399487619618753359356, −8.341275212548067016845372469550, −8.341275212548067016845372469550, −7.60583694658959035837478078570, −7.60583694658959035837478078570, −6.37318226789872439689553131647, −6.37318226789872439689553131647, −5.48198608682619476325367860976, −5.48198608682619476325367860976, −4.24977516365535388494352784618, −4.24977516365535388494352784618, −3.07984083787626508123182523604, −3.07984083787626508123182523604, −1.36018793518762280095521620239, −1.36018793518762280095521620239,
1.36018793518762280095521620239, 1.36018793518762280095521620239, 3.07984083787626508123182523604, 3.07984083787626508123182523604, 4.24977516365535388494352784618, 4.24977516365535388494352784618, 5.48198608682619476325367860976, 5.48198608682619476325367860976, 6.37318226789872439689553131647, 6.37318226789872439689553131647, 7.60583694658959035837478078570, 7.60583694658959035837478078570, 8.341275212548067016845372469550, 8.341275212548067016845372469550, 9.387437094399487619618753359356, 9.387437094399487619618753359356, 10.22208553622788501090350672882, 10.22208553622788501090350672882, 11.38842166918163472638466197964, 11.38842166918163472638466197964, 11.70175137554538908977854303538, 11.70175137554538908977854303538, 13.03049201961870670350835369127, 13.03049201961870670350835369127, 13.64964189773580716903063683793
