| L(s) = 1 | − 5-s + 7-s + 2·11-s + 5·13-s − 8·17-s − 10·19-s + 2·23-s − 10·29-s + 8·31-s − 35-s − 6·37-s − 6·41-s − 4·43-s + 8·47-s + 7·49-s + 12·53-s − 2·55-s + 4·59-s + 5·61-s − 5·65-s + 7·67-s + 12·71-s − 18·73-s + 2·77-s − 3·79-s − 2·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s + 1.38·13-s − 1.94·17-s − 2.29·19-s + 0.417·23-s − 1.85·29-s + 1.43·31-s − 0.169·35-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 49-s + 1.64·53-s − 0.269·55-s + 0.520·59-s + 0.640·61-s − 0.620·65-s + 0.855·67-s + 1.42·71-s − 2.10·73-s + 0.227·77-s − 0.337·79-s − 0.219·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.467471502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467471502\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.722733822840764520888617058245, −8.398943981190422521671878335549, −8.376409790926206178647905351340, −7.891678216095794416998142827967, −7.04459811558493876626876272795, −6.93410475560692207718248810314, −6.81546060984119447939551891464, −6.24108497250037474737755153225, −5.80410108684647222389025687379, −5.57640740447330907711964446747, −4.89094855094810274813565626061, −4.45418684630574356493718713006, −4.11724954176897725403269792128, −3.89230188942959940168501655324, −3.48797558544588749895944457065, −2.72658674335477858753098944222, −2.10983549793916148648753275501, −1.96939720214591320692318070281, −1.18961209687811799822285303567, −0.38934661454709129745777601041,
0.38934661454709129745777601041, 1.18961209687811799822285303567, 1.96939720214591320692318070281, 2.10983549793916148648753275501, 2.72658674335477858753098944222, 3.48797558544588749895944457065, 3.89230188942959940168501655324, 4.11724954176897725403269792128, 4.45418684630574356493718713006, 4.89094855094810274813565626061, 5.57640740447330907711964446747, 5.80410108684647222389025687379, 6.24108497250037474737755153225, 6.81546060984119447939551891464, 6.93410475560692207718248810314, 7.04459811558493876626876272795, 7.891678216095794416998142827967, 8.376409790926206178647905351340, 8.398943981190422521671878335549, 8.722733822840764520888617058245
