| L(s) = 1 | − 9-s + 2·11-s − 4·19-s + 16·31-s + 10·49-s + 24·59-s + 4·61-s + 20·79-s + 81-s + 12·89-s − 2·99-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 4·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.603·11-s − 0.917·19-s + 2.87·31-s + 10/7·49-s + 3.12·59-s + 0.512·61-s + 2.25·79-s + 1/9·81-s + 1.27·89-s − 0.201·99-s + 1.91·109-s + 3/11·121-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 + 1.69·169-s + 0.305·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.213814817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.213814817\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.749883580812812006165559593518, −8.417993999122107958994227919729, −8.023050242787469619107936975439, −8.016855414645358646631839727317, −7.14416415361375159744492169209, −6.98800988925809211714705530259, −6.48555567801811800575149277367, −6.38428292344898787151856786810, −5.65838985957645652872963388491, −5.63342819774202636020641213180, −4.88393883460047544735807971026, −4.60515657598041915143825482996, −4.15989753722274716419564127055, −3.80083768045080560135210661035, −3.26373092169547735242941718306, −2.79981776618904306742564714308, −2.16607851583041693523935364165, −2.03063919585748110170271746478, −0.839488565553105783029531608823, −0.77154781730770278048086316344,
0.77154781730770278048086316344, 0.839488565553105783029531608823, 2.03063919585748110170271746478, 2.16607851583041693523935364165, 2.79981776618904306742564714308, 3.26373092169547735242941718306, 3.80083768045080560135210661035, 4.15989753722274716419564127055, 4.60515657598041915143825482996, 4.88393883460047544735807971026, 5.63342819774202636020641213180, 5.65838985957645652872963388491, 6.38428292344898787151856786810, 6.48555567801811800575149277367, 6.98800988925809211714705530259, 7.14416415361375159744492169209, 8.016855414645358646631839727317, 8.023050242787469619107936975439, 8.417993999122107958994227919729, 8.749883580812812006165559593518
