| L(s) = 1 | + 4·5-s + 4·7-s + 4·11-s + 4·17-s − 8·23-s + 4·25-s + 4·29-s + 12·31-s + 16·35-s + 8·37-s + 12·41-s + 8·43-s − 8·47-s + 4·53-s + 16·55-s − 8·59-s + 8·61-s + 16·67-s − 24·71-s + 8·73-s + 16·77-s + 20·79-s − 4·83-s + 16·85-s − 4·89-s + 4·97-s − 4·101-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s + 1.20·11-s + 0.970·17-s − 1.66·23-s + 4/5·25-s + 0.742·29-s + 2.15·31-s + 2.70·35-s + 1.31·37-s + 1.87·41-s + 1.21·43-s − 1.16·47-s + 0.549·53-s + 2.15·55-s − 1.04·59-s + 1.02·61-s + 1.95·67-s − 2.84·71-s + 0.936·73-s + 1.82·77-s + 2.25·79-s − 0.439·83-s + 1.73·85-s − 0.423·89-s + 0.406·97-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.872712332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.872712332\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 240 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.331855736467897710675438440363, −8.115986525583857272656340209959, −7.76286562195232263396088706111, −7.73767693179721289177421371333, −6.91975511187405688868801555274, −6.51938349739942481526667437844, −6.15238525488398953993879243620, −6.10361659804626254678279783841, −5.61020677836677848325583594990, −5.30034851041620303420082673160, −4.64301621794781698873241075286, −4.61692563488086584660718316133, −3.96683080443626455630115761002, −3.78443940346819520216077468930, −2.75837890263118224056386816817, −2.70841013164296153844322149236, −1.93723054310808572352251286324, −1.83619650703641114579002383234, −1.08709898077617771084790018006, −0.918619376833668659725951051311,
0.918619376833668659725951051311, 1.08709898077617771084790018006, 1.83619650703641114579002383234, 1.93723054310808572352251286324, 2.70841013164296153844322149236, 2.75837890263118224056386816817, 3.78443940346819520216077468930, 3.96683080443626455630115761002, 4.61692563488086584660718316133, 4.64301621794781698873241075286, 5.30034851041620303420082673160, 5.61020677836677848325583594990, 6.10361659804626254678279783841, 6.15238525488398953993879243620, 6.51938349739942481526667437844, 6.91975511187405688868801555274, 7.73767693179721289177421371333, 7.76286562195232263396088706111, 8.115986525583857272656340209959, 8.331855736467897710675438440363
