| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.749168748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749168748\) |
| \(L(\frac{3}{2})\) |
|
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
−9.697088823818481711701633072257, −9.272173387758320515469722329583, −9.017677189013623707133571075219, −8.595105853823862202606009325874, −8.204675737945985483454912992224, −7.65984502706950344450940204942, −7.51063017902087504211867564221, −6.66561719375102986601028406148, −6.50816979297899993781651137106, −6.28598407300574551655310136465, −5.73249939628901902563791392083, −5.07063496362356523054172266486, −4.52035633040959335322798926595, −4.22386622710022987780700289512, −4.03881938396490623702544436908, −3.03642885803833412542292928486, −2.82801833982336992243392563522, −1.74823676947384655770563592336, −1.73907920124756321652326457018, −0.53235295144991596471459908194,
0.53235295144991596471459908194, 1.73907920124756321652326457018, 1.74823676947384655770563592336, 2.82801833982336992243392563522, 3.03642885803833412542292928486, 4.03881938396490623702544436908, 4.22386622710022987780700289512, 4.52035633040959335322798926595, 5.07063496362356523054172266486, 5.73249939628901902563791392083, 6.28598407300574551655310136465, 6.50816979297899993781651137106, 6.66561719375102986601028406148, 7.51063017902087504211867564221, 7.65984502706950344450940204942, 8.204675737945985483454912992224, 8.595105853823862202606009325874, 9.017677189013623707133571075219, 9.272173387758320515469722329583, 9.697088823818481711701633072257
