| L(s) = 1 | + 5·9-s − 10·11-s + 10·19-s − 8·29-s − 20·31-s + 10·41-s + 10·49-s − 20·61-s + 20·79-s + 16·81-s + 18·89-s − 50·99-s + 4·101-s − 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 50·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 3.01·11-s + 2.29·19-s − 1.48·29-s − 3.59·31-s + 1.56·41-s + 10/7·49-s − 2.56·61-s + 2.25·79-s + 16/9·81-s + 1.90·89-s − 5.02·99-s + 0.398·101-s − 1.91·109-s + 4.81·121-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 + 2·169-s + 3.82·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545648143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545648143\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.69967645107177538316512433065, −10.11694107985850541027209041619, −9.527619566537075065745215194405, −9.217931996999662089968219252922, −9.168831128370293034551050742766, −7.975982946575490956065825743494, −7.80842805386634817278174003932, −7.40783758239711510963329526499, −7.40069371744290710077357183824, −6.80654719703062744747413362983, −5.76220579796162224746435553519, −5.46153784926225307614962435720, −5.39548034411452532578575387076, −4.67557328665462077938718109065, −4.16177593302263232414571584845, −3.37279394231353940314904497018, −3.14216657482462393353469061253, −2.15701418988363723904694284861, −1.81327239186058996585292624903, −0.61416970654333876974146005537,
0.61416970654333876974146005537, 1.81327239186058996585292624903, 2.15701418988363723904694284861, 3.14216657482462393353469061253, 3.37279394231353940314904497018, 4.16177593302263232414571584845, 4.67557328665462077938718109065, 5.39548034411452532578575387076, 5.46153784926225307614962435720, 5.76220579796162224746435553519, 6.80654719703062744747413362983, 7.40069371744290710077357183824, 7.40783758239711510963329526499, 7.80842805386634817278174003932, 7.975982946575490956065825743494, 9.168831128370293034551050742766, 9.217931996999662089968219252922, 9.527619566537075065745215194405, 10.11694107985850541027209041619, 10.69967645107177538316512433065
