L(s) = 1 | − 4·7-s + 4·17-s − 4·23-s − 25-s − 8·31-s − 8·41-s + 4·47-s − 8·71-s − 4·73-s − 9·81-s − 4·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 16·119-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 0.970·17-s − 0.834·23-s − 1/5·25-s − 1.43·31-s − 1.24·41-s + 0.583·47-s − 0.949·71-s − 0.468·73-s − 81-s − 0.423·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.46·119-s + 2/11·121-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 + 0.0798·157-s + 1.26·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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
−14.2151067192, −13.8039112904, −13.2899048446, −12.9128056800, −12.6158677963, −12.0671157476, −11.7615203711, −11.1848198677, −10.5947354095, −10.0508482539, −9.94138507393, −9.35783223000, −8.90142020079, −8.40167910522, −7.67834979289, −7.39129337813, −6.69484144053, −6.31722754152, −5.70933179667, −5.32752819203, −4.45634499647, −3.68781758863, −3.36447939541, −2.60553505577, −1.57217239405, 0,
1.57217239405, 2.60553505577, 3.36447939541, 3.68781758863, 4.45634499647, 5.32752819203, 5.70933179667, 6.31722754152, 6.69484144053, 7.39129337813, 7.67834979289, 8.40167910522, 8.90142020079, 9.35783223000, 9.94138507393, 10.0508482539, 10.5947354095, 11.1848198677, 11.7615203711, 12.0671157476, 12.6158677963, 12.9128056800, 13.2899048446, 13.8039112904, 14.2151067192