| L(s) = 1 | + 3.04·3-s + 5-s + 3.43·7-s + 6.26·9-s − 0.254·11-s − 2.66·13-s + 3.04·15-s + 0.449·17-s − 4.69·19-s + 10.4·21-s − 0.415·23-s + 25-s + 9.93·27-s + 29-s − 5.51·31-s − 0.774·33-s + 3.43·35-s − 7.29·37-s − 8.12·39-s − 4.86·41-s + 2.44·43-s + 6.26·45-s − 12.6·47-s + 4.77·49-s + 1.36·51-s + 8.60·53-s − 0.254·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 0.447·5-s + 1.29·7-s + 2.08·9-s − 0.0767·11-s − 0.740·13-s + 0.785·15-s + 0.109·17-s − 1.07·19-s + 2.27·21-s − 0.0865·23-s + 0.200·25-s + 1.91·27-s + 0.185·29-s − 0.991·31-s − 0.134·33-s + 0.579·35-s − 1.19·37-s − 1.30·39-s − 0.759·41-s + 0.373·43-s + 0.933·45-s − 1.84·47-s + 0.681·49-s + 0.191·51-s + 1.18·53-s − 0.0343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.516875428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.516875428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 0.254T + 11T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 0.449T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 0.415T + 23T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 - 4.16T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 0.185T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 9.24T + 97T^{2} \) |
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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
−9.681295010611156202041081217687, −8.722020755307850858941598180697, −8.356329709706303822011415875665, −7.54633122895044298946965410130, −6.78256103649841922287883016233, −5.30212568897666243472800531294, −4.47354503152814597240065825288, −3.46874519434779586050589746420, −2.29452224919892585739625483141, −1.71989866536344280379084609593,
1.71989866536344280379084609593, 2.29452224919892585739625483141, 3.46874519434779586050589746420, 4.47354503152814597240065825288, 5.30212568897666243472800531294, 6.78256103649841922287883016233, 7.54633122895044298946965410130, 8.356329709706303822011415875665, 8.722020755307850858941598180697, 9.681295010611156202041081217687
