| L(s) = 1 | + 7-s + 9-s − 4·11-s + 8·23-s + 25-s + 12·29-s + 20·37-s + 24·43-s + 49-s + 63-s − 24·67-s − 20·71-s − 4·77-s + 81-s − 4·99-s + 8·107-s − 28·109-s + 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·23-s + 1/5·25-s + 2.22·29-s + 3.28·37-s + 3.65·43-s + 1/7·49-s + 0.125·63-s − 2.93·67-s − 2.37·71-s − 0.455·77-s + 1/9·81-s − 0.402·99-s + 0.773·107-s − 2.68·109-s + 0.752·113-s − 0.909·121-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 + 0.630·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.478972397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478972397\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.82067649776875631788978756567, −7.51325888882796800062561076037, −7.46985318735628348223218545659, −6.62104364944396067563104383367, −6.31888690467862628757406619126, −5.67017038367597175775538292471, −5.51088025071477982042675051210, −4.60737339403970738856758018331, −4.51008988119235510509832391491, −4.18636216199252309981165234200, −3.00191084542596222591694033423, −2.80785568081983867387706785012, −2.44981952348987740648309403533, −1.28136430783138726696385588581, −0.798786744184690487080656808366,
0.798786744184690487080656808366, 1.28136430783138726696385588581, 2.44981952348987740648309403533, 2.80785568081983867387706785012, 3.00191084542596222591694033423, 4.18636216199252309981165234200, 4.51008988119235510509832391491, 4.60737339403970738856758018331, 5.51088025071477982042675051210, 5.67017038367597175775538292471, 6.31888690467862628757406619126, 6.62104364944396067563104383367, 7.46985318735628348223218545659, 7.51325888882796800062561076037, 7.82067649776875631788978756567
