| L(s) = 1 | − 2-s − 1.91·3-s + 4-s − 4.19·5-s + 1.91·6-s − 8-s + 0.658·9-s + 4.19·10-s − 1.42·11-s − 1.91·12-s + 2.65·13-s + 8.01·15-s + 16-s − 1.68·17-s − 0.658·18-s − 0.426·19-s − 4.19·20-s + 1.42·22-s − 4.21·23-s + 1.91·24-s + 12.5·25-s − 2.65·26-s + 4.47·27-s + 7.40·29-s − 8.01·30-s − 9.93·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s − 1.87·5-s + 0.780·6-s − 0.353·8-s + 0.219·9-s + 1.32·10-s − 0.430·11-s − 0.552·12-s + 0.735·13-s + 2.06·15-s + 0.250·16-s − 0.407·17-s − 0.155·18-s − 0.0978·19-s − 0.937·20-s + 0.304·22-s − 0.879·23-s + 0.390·24-s + 2.51·25-s − 0.520·26-s + 0.861·27-s + 1.37·29-s − 1.46·30-s − 1.78·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1727399346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1727399346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 11 | \( 1 + 1.42T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 0.426T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 7.40T + 29T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 0.791T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 0.426T + 73T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−7.87964295841255121496838886959, −7.31739336865717429710629862855, −6.47437271594144697003145386227, −6.02940269582048452651981901111, −4.97927685176430751057578611598, −4.38777318893342171685651331284, −3.55802549294354702367675848137, −2.75484115986771037084986267071, −1.30714941844917299913623271362, −0.26712498468233387940158984078,
0.26712498468233387940158984078, 1.30714941844917299913623271362, 2.75484115986771037084986267071, 3.55802549294354702367675848137, 4.38777318893342171685651331284, 4.97927685176430751057578611598, 6.02940269582048452651981901111, 6.47437271594144697003145386227, 7.31739336865717429710629862855, 7.87964295841255121496838886959
