| L(s) = 1 | + 2·9-s − 2·11-s + 12·19-s + 6·29-s + 4·41-s − 49-s + 16·59-s − 20·61-s + 14·71-s − 18·79-s − 5·81-s + 8·89-s − 4·99-s − 28·101-s + 22·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 24·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.603·11-s + 2.75·19-s + 1.11·29-s + 0.624·41-s − 1/7·49-s + 2.08·59-s − 2.56·61-s + 1.66·71-s − 2.02·79-s − 5/9·81-s + 0.847·89-s − 0.402·99-s − 2.78·101-s + 2.10·109-s − 1.72·121-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 + 0.769·169-s + 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.955809567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.955809567\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.305269082379328462856966552538, −8.470406925835257224876919855044, −8.271173931353548175343207466521, −7.75147914060260894247591903181, −7.49055053920893574422695295191, −7.22005348022164819018215937340, −6.73457926145965566675673375587, −6.45339982788102872040577029105, −5.66599678739973971892782164914, −5.62593631804495930900112961848, −5.17070944203388447469133486069, −4.63161366664251013192301746204, −4.41292893251805478722163867192, −3.73385080153482245591792724714, −3.34485876795042671043189042136, −2.83559990731355648314662776524, −2.55084998666211776549106254303, −1.68519892909677200909587451682, −1.20961378144737726799388679502, −0.61781531551502741500821885337,
0.61781531551502741500821885337, 1.20961378144737726799388679502, 1.68519892909677200909587451682, 2.55084998666211776549106254303, 2.83559990731355648314662776524, 3.34485876795042671043189042136, 3.73385080153482245591792724714, 4.41292893251805478722163867192, 4.63161366664251013192301746204, 5.17070944203388447469133486069, 5.62593631804495930900112961848, 5.66599678739973971892782164914, 6.45339982788102872040577029105, 6.73457926145965566675673375587, 7.22005348022164819018215937340, 7.49055053920893574422695295191, 7.75147914060260894247591903181, 8.271173931353548175343207466521, 8.470406925835257224876919855044, 9.305269082379328462856966552538
