| L(s) = 1 | + 2.37·5-s + 2·7-s − 2.37·11-s − 0.372·13-s + 17-s + 6.37·19-s + 4.37·23-s + 0.627·25-s − 4.74·29-s − 2·31-s + 4.74·35-s + 4·37-s − 3.62·41-s + 11.1·43-s + 4·47-s − 3·49-s − 6.74·53-s − 5.62·55-s + 0.744·59-s + 8.74·61-s − 0.883·65-s + 4·67-s + 1.25·71-s + 14.7·73-s − 4.74·77-s − 2·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s + 0.755·7-s − 0.715·11-s − 0.103·13-s + 0.242·17-s + 1.46·19-s + 0.911·23-s + 0.125·25-s − 0.881·29-s − 0.359·31-s + 0.801·35-s + 0.657·37-s − 0.566·41-s + 1.69·43-s + 0.583·47-s − 0.428·49-s − 0.926·53-s − 0.758·55-s + 0.0969·59-s + 1.11·61-s − 0.109·65-s + 0.488·67-s + 0.148·71-s + 1.72·73-s − 0.540·77-s − 0.225·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.754387199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754387199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 3.62T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 6.74T + 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.117592689300812357640999148276, −7.62055256052759117710335953694, −6.88560652284055589642916146169, −5.86984933770762510253291470084, −5.36648764525211078665313450497, −4.83882275063491798827251642425, −3.67927091955995892445940012300, −2.71610556825917031707046247163, −1.93266109125584714577750789328, −0.954320009094335107604431114378,
0.954320009094335107604431114378, 1.93266109125584714577750789328, 2.71610556825917031707046247163, 3.67927091955995892445940012300, 4.83882275063491798827251642425, 5.36648764525211078665313450497, 5.86984933770762510253291470084, 6.88560652284055589642916146169, 7.62055256052759117710335953694, 8.117592689300812357640999148276
