| L(s) = 1 | − 9-s + 8·11-s + 8·19-s − 12·29-s + 16·31-s − 12·41-s + 14·49-s − 8·59-s − 4·61-s + 16·71-s + 16·79-s + 81-s + 12·89-s − 8·99-s − 36·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s + 1.83·19-s − 2.22·29-s + 2.87·31-s − 1.87·41-s + 2·49-s − 1.04·59-s − 0.512·61-s + 1.89·71-s + 1.80·79-s + 1/9·81-s + 1.27·89-s − 0.804·99-s − 3.58·101-s + 0.383·109-s + 2.36·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273403407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273403407\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.84689869069326518718596802165, −10.55040768262254683950084753932, −9.722755076682735790108006807249, −9.555110999509083129577208498245, −9.319980256554947494896591516093, −8.744876854152440905642778675731, −8.305801328331706763494157577162, −7.81492004770371017925162683408, −7.22689213984221627371390229595, −6.89625889003198805119507742819, −6.25742097909589294180313204260, −6.12870744714571890620967147062, −5.25755492369751829160969529345, −5.00001355591556398908592636892, −4.13023788952651458450579076642, −3.75594847300271695208947856097, −3.29839074888651189590309568332, −2.50167583361520930081768631471, −1.55789354773542048252640331200, −0.959303705450604034011833009324,
0.959303705450604034011833009324, 1.55789354773542048252640331200, 2.50167583361520930081768631471, 3.29839074888651189590309568332, 3.75594847300271695208947856097, 4.13023788952651458450579076642, 5.00001355591556398908592636892, 5.25755492369751829160969529345, 6.12870744714571890620967147062, 6.25742097909589294180313204260, 6.89625889003198805119507742819, 7.22689213984221627371390229595, 7.81492004770371017925162683408, 8.305801328331706763494157577162, 8.744876854152440905642778675731, 9.319980256554947494896591516093, 9.555110999509083129577208498245, 9.722755076682735790108006807249, 10.55040768262254683950084753932, 10.84689869069326518718596802165
