L(s) = 1 | − 5-s − 3·9-s + 8·19-s + 25-s − 12·29-s + 3·45-s − 14·49-s + 9·81-s − 8·95-s − 12·101-s + 10·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s − 24·171-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 9-s + 1.83·19-s + 1/5·25-s − 2.22·29-s + 0.447·45-s − 2·49-s + 81-s − 0.820·95-s − 1.19·101-s + 0.909·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s − 1.83·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.594779585562341701557044549056, −8.107214067031437191645231100881, −7.70375715931405159748387359255, −7.31645525203616759051435637813, −6.83379913931150784842540267425, −6.11625298589384198997194601457, −5.74179788640160563684999647273, −5.19324962455824748504258393594, −4.85007630846265541527083913236, −3.96676451549244119501752011568, −3.44217811334306033362634981695, −3.05987293362605700214423169345, −2.22840581896462101386015856132, −1.29774637819727219192126572593, 0,
1.29774637819727219192126572593, 2.22840581896462101386015856132, 3.05987293362605700214423169345, 3.44217811334306033362634981695, 3.96676451549244119501752011568, 4.85007630846265541527083913236, 5.19324962455824748504258393594, 5.74179788640160563684999647273, 6.11625298589384198997194601457, 6.83379913931150784842540267425, 7.31645525203616759051435637813, 7.70375715931405159748387359255, 8.107214067031437191645231100881, 8.594779585562341701557044549056