| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 12·17-s + 8·23-s − 25-s − 4·27-s − 20·29-s − 3·36-s + 8·43-s − 2·48-s + 14·49-s − 24·51-s − 12·53-s + 12·61-s − 64-s − 12·68-s − 16·69-s + 2·75-s + 5·81-s + 40·87-s − 8·92-s + 100-s − 12·101-s − 24·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 2.91·17-s + 1.66·23-s − 1/5·25-s − 0.769·27-s − 3.71·29-s − 1/2·36-s + 1.21·43-s − 0.288·48-s + 2·49-s − 3.36·51-s − 1.64·53-s + 1.53·61-s − 1/8·64-s − 1.45·68-s − 1.92·69-s + 0.230·75-s + 5/9·81-s + 4.28·87-s − 0.834·92-s + 1/10·100-s − 1.19·101-s − 2.36·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.915307321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915307321\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.198135375919331840457778272343, −8.029691840275287450711701158471, −7.52591359610023929216041127757, −7.41234312493628233856416780872, −6.99943850519943445645605652065, −6.73375902614860940196976603363, −5.95196049065761391536447184020, −5.64863278600151110438589543820, −5.56760850127032610751125474591, −5.42196958548121132058753421517, −4.83339163629884907695614539457, −4.39061178178631468494967381001, −3.87087845375108420712881376024, −3.67494754847133786121739308508, −3.17229936165954429728502411040, −2.77687199364548383130086485012, −1.84783341576565889441710241662, −1.59902414252681289599772025813, −0.810313891126520502397480983647, −0.58010209589135118107773433150,
0.58010209589135118107773433150, 0.810313891126520502397480983647, 1.59902414252681289599772025813, 1.84783341576565889441710241662, 2.77687199364548383130086485012, 3.17229936165954429728502411040, 3.67494754847133786121739308508, 3.87087845375108420712881376024, 4.39061178178631468494967381001, 4.83339163629884907695614539457, 5.42196958548121132058753421517, 5.56760850127032610751125474591, 5.64863278600151110438589543820, 5.95196049065761391536447184020, 6.73375902614860940196976603363, 6.99943850519943445645605652065, 7.41234312493628233856416780872, 7.52591359610023929216041127757, 8.029691840275287450711701158471, 8.198135375919331840457778272343
