| L(s) = 1 | + 8·7-s − 2·9-s − 4·17-s − 8·23-s − 6·25-s − 12·41-s + 16·47-s + 34·49-s − 16·63-s − 24·71-s + 28·73-s + 16·79-s − 5·81-s − 4·89-s − 4·97-s + 8·103-s + 4·113-s − 32·119-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2/3·9-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 1.87·41-s + 2.33·47-s + 34/7·49-s − 2.01·63-s − 2.84·71-s + 3.27·73-s + 1.80·79-s − 5/9·81-s − 0.423·89-s − 0.406·97-s + 0.788·103-s + 0.376·113-s − 2.93·119-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 + 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.334302111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334302111\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.28853583105345723449818824184, −10.57265847830629711938727340297, −10.22941768705266383849925633731, −9.215939702261382322141825805489, −8.728992519223720851294830795286, −8.098681382024412731658064042949, −7.984207807149970007419508597610, −7.33738117792255002879146989083, −6.39871498618546548325782647590, −5.59100833138318705450052051530, −5.16542238567969836254694300670, −4.43324404972048918127214240916, −3.91058908136761919842529082491, −2.30676446591740616355718333113, −1.73703685485230483844290438322,
1.73703685485230483844290438322, 2.30676446591740616355718333113, 3.91058908136761919842529082491, 4.43324404972048918127214240916, 5.16542238567969836254694300670, 5.59100833138318705450052051530, 6.39871498618546548325782647590, 7.33738117792255002879146989083, 7.984207807149970007419508597610, 8.098681382024412731658064042949, 8.728992519223720851294830795286, 9.215939702261382322141825805489, 10.22941768705266383849925633731, 10.57265847830629711938727340297, 11.28853583105345723449818824184
