| L(s) = 1 | + 0.577·5-s + 2.32·7-s − 11-s − 13-s + 4.24·17-s + 1.26·19-s − 3.28·23-s − 4.66·25-s + 10.0·29-s + 6.24·31-s + 1.34·35-s − 9.82·37-s + 3.76·41-s − 1.64·43-s + 7.17·47-s − 1.59·49-s + 2.94·53-s − 0.577·55-s + 0.161·59-s + 4.92·61-s − 0.577·65-s + 13.7·67-s − 4.97·71-s + 6.63·73-s − 2.32·77-s − 5.30·79-s + 0.0291·83-s + ⋯ |
| L(s) = 1 | + 0.258·5-s + 0.878·7-s − 0.301·11-s − 0.277·13-s + 1.02·17-s + 0.290·19-s − 0.685·23-s − 0.933·25-s + 1.85·29-s + 1.12·31-s + 0.226·35-s − 1.61·37-s + 0.588·41-s − 0.251·43-s + 1.04·47-s − 0.228·49-s + 0.403·53-s − 0.0778·55-s + 0.0209·59-s + 0.630·61-s − 0.0716·65-s + 1.67·67-s − 0.590·71-s + 0.776·73-s − 0.264·77-s − 0.596·79-s + 0.00319·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354483288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354483288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.577T + 5T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 - 0.161T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 4.97T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 + 5.30T + 79T^{2} \) |
| 83 | \( 1 - 0.0291T + 83T^{2} \) |
| 89 | \( 1 - 2.17T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.190852776016354762353139206431, −7.63365558895024799937474046407, −6.81979697142228198857766211212, −5.96165492907109071916116916074, −5.27869317703775534466350398867, −4.65120178557359319667500100569, −3.74604455512744057795015848073, −2.76052574339211213888589331451, −1.89603793070050302033777448911, −0.860842605953004788373581020875,
0.860842605953004788373581020875, 1.89603793070050302033777448911, 2.76052574339211213888589331451, 3.74604455512744057795015848073, 4.65120178557359319667500100569, 5.27869317703775534466350398867, 5.96165492907109071916116916074, 6.81979697142228198857766211212, 7.63365558895024799937474046407, 8.190852776016354762353139206431
