| L(s) = 1 | + 32·11-s − 48·17-s − 48·19-s + 34·25-s + 16·41-s + 112·43-s + 82·49-s − 64·59-s − 160·67-s + 132·73-s + 32·83-s − 288·89-s + 188·97-s − 192·107-s + 160·113-s + 526·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 334·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2.90·11-s − 2.82·17-s − 2.52·19-s + 1.35·25-s + 0.390·41-s + 2.60·43-s + 1.67·49-s − 1.08·59-s − 2.38·67-s + 1.80·73-s + 0.385·83-s − 3.23·89-s + 1.93·97-s − 1.79·107-s + 1.41·113-s + 4.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.97·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.759228643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.759228643\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 334 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 254 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 782 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2414 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3394 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4322 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 3598 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 6302 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12082 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−8.998489168823996628911742508019, −8.831975550804144773219545731160, −8.409270111647656166532021588241, −8.040283867698019477288637202717, −7.15015065767248024755014650932, −6.97585118251404803111939327429, −6.76997426356474344258402075408, −6.36866694901376427880845073840, −5.93760239488880150697762706674, −5.81945315304485823533833179492, −4.71267407267209639986515282173, −4.38819597122959207317778831029, −4.23670546566554669834971560987, −4.07947724261130153921376264417, −3.29709819525905895785430123336, −2.63753659459538533613567270449, −2.11840818008686778674673604188, −1.81329309477642910243067714140, −1.04417980054263577202256055555, −0.44801027293771661854589471607,
0.44801027293771661854589471607, 1.04417980054263577202256055555, 1.81329309477642910243067714140, 2.11840818008686778674673604188, 2.63753659459538533613567270449, 3.29709819525905895785430123336, 4.07947724261130153921376264417, 4.23670546566554669834971560987, 4.38819597122959207317778831029, 4.71267407267209639986515282173, 5.81945315304485823533833179492, 5.93760239488880150697762706674, 6.36866694901376427880845073840, 6.76997426356474344258402075408, 6.97585118251404803111939327429, 7.15015065767248024755014650932, 8.040283867698019477288637202717, 8.409270111647656166532021588241, 8.831975550804144773219545731160, 8.998489168823996628911742508019
