| L(s) = 1 | + 3.17·3-s + 0.276·5-s + 7-s + 7.05·9-s + 1.52·11-s + 5.05·13-s + 0.875·15-s + 4.14·19-s + 3.17·21-s + 1.24·23-s − 4.92·25-s + 12.8·27-s + 7.81·29-s + 3.55·31-s + 4.82·33-s + 0.276·35-s − 12.0·37-s + 16.0·39-s + 3.43·41-s − 3.55·43-s + 1.94·45-s − 0.580·47-s + 49-s − 3.04·53-s + 0.419·55-s + 13.1·57-s − 10.4·59-s + ⋯ |
| L(s) = 1 | + 1.83·3-s + 0.123·5-s + 0.377·7-s + 2.35·9-s + 0.458·11-s + 1.40·13-s + 0.226·15-s + 0.952·19-s + 0.692·21-s + 0.259·23-s − 0.984·25-s + 2.47·27-s + 1.45·29-s + 0.638·31-s + 0.839·33-s + 0.0466·35-s − 1.98·37-s + 2.56·39-s + 0.536·41-s − 0.542·43-s + 0.290·45-s − 0.0846·47-s + 0.142·49-s − 0.418·53-s + 0.0566·55-s + 1.74·57-s − 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.575570220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.575570220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 - 0.276T + 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 - 7.81T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 3.55T + 43T^{2} \) |
| 47 | \( 1 + 0.580T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 - 0.470T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 5.00T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.123210535142268233462474209308, −7.28427497681969537078720803169, −6.66507853590357177598123906041, −5.81915977302943571798671811423, −4.76586259625611521683578256013, −4.06723970234652194480522616521, −3.35939750539949306957636331300, −2.85403707811585648511746192569, −1.73861079795688961900140811198, −1.24379586853704035074385937339,
1.24379586853704035074385937339, 1.73861079795688961900140811198, 2.85403707811585648511746192569, 3.35939750539949306957636331300, 4.06723970234652194480522616521, 4.76586259625611521683578256013, 5.81915977302943571798671811423, 6.66507853590357177598123906041, 7.28427497681969537078720803169, 8.123210535142268233462474209308
