| L(s) = 1 | − 3·9-s + 4·13-s + 20·37-s + 2·49-s + 28·61-s − 20·73-s + 9·81-s + 28·97-s − 4·109-s − 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 3.28·37-s + 2/7·49-s + 3.58·61-s − 2.34·73-s + 81-s + 2.84·97-s − 0.383·109-s − 1.10·117-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043246287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043246287\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.806252096077230051549615030773, −9.648419446908427041658540004991, −9.004961524472074192758532613896, −8.762550242819512383214775020218, −8.278234613837785937109730992817, −8.109096388454995116136049978616, −7.39194679382306185360896595271, −7.26195777582282358024226536639, −6.31757887149443789052477207507, −6.27345544268238272974020158098, −5.87738937373433097079196097912, −5.32281012845077203772794084647, −4.88789315724740340779684229735, −4.23829177968066935399154452744, −3.83186428156488392870057782850, −3.36105338282589031889763531663, −2.55365850711334460748557892017, −2.43542790409807925484042560071, −1.33600478664384813188412092758, −0.66784532134024500454975933619,
0.66784532134024500454975933619, 1.33600478664384813188412092758, 2.43542790409807925484042560071, 2.55365850711334460748557892017, 3.36105338282589031889763531663, 3.83186428156488392870057782850, 4.23829177968066935399154452744, 4.88789315724740340779684229735, 5.32281012845077203772794084647, 5.87738937373433097079196097912, 6.27345544268238272974020158098, 6.31757887149443789052477207507, 7.26195777582282358024226536639, 7.39194679382306185360896595271, 8.109096388454995116136049978616, 8.278234613837785937109730992817, 8.762550242819512383214775020218, 9.004961524472074192758532613896, 9.648419446908427041658540004991, 9.806252096077230051549615030773
