| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s − 14·19-s − 10·29-s + 8·31-s + 36-s − 10·41-s − 8·44-s − 2·49-s + 12·59-s + 8·61-s − 64-s − 30·71-s + 14·76-s − 14·79-s + 81-s − 28·89-s − 8·99-s − 20·101-s + 22·109-s + 10·116-s + 26·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 3.21·19-s − 1.85·29-s + 1.43·31-s + 1/6·36-s − 1.56·41-s − 1.20·44-s − 2/7·49-s + 1.56·59-s + 1.02·61-s − 1/8·64-s − 3.56·71-s + 1.60·76-s − 1.57·79-s + 1/9·81-s − 2.96·89-s − 0.804·99-s − 1.99·101-s + 2.10·109-s + 0.928·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172291159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172291159\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.384106465165423691845979452372, −8.739592408247083859082926066477, −8.670231061257909564688716669936, −8.459272151221624133241446892714, −8.070356942631030067294721460575, −7.17679489994874949518618585846, −6.99470142407571856554613755995, −6.68822946218368444751011820651, −6.11767895961085587848521711702, −5.95894365267097533067131892314, −5.49039042451298610184716050483, −4.73570751545750565196748619118, −4.22718074088166415958433641186, −4.20544418149630622183628517188, −3.75309334582671404560127280816, −3.12536449962335521660315023991, −2.48870016853718065610658044961, −1.70879922553749039211126270533, −1.53644023203170128004644993958, −0.39435880015114462270059385706,
0.39435880015114462270059385706, 1.53644023203170128004644993958, 1.70879922553749039211126270533, 2.48870016853718065610658044961, 3.12536449962335521660315023991, 3.75309334582671404560127280816, 4.20544418149630622183628517188, 4.22718074088166415958433641186, 4.73570751545750565196748619118, 5.49039042451298610184716050483, 5.95894365267097533067131892314, 6.11767895961085587848521711702, 6.68822946218368444751011820651, 6.99470142407571856554613755995, 7.17679489994874949518618585846, 8.070356942631030067294721460575, 8.459272151221624133241446892714, 8.670231061257909564688716669936, 8.739592408247083859082926066477, 9.384106465165423691845979452372
