| L(s) = 1 | − 13.5·7-s + 9·9-s − 21.5·11-s + 25·25-s + 34.4·31-s + 17.0·37-s − 51.2·41-s + 70.8·43-s + 30.8·47-s + 134.·49-s + 74.4·59-s − 119.·61-s − 121.·63-s + 73·73-s + 291.·77-s + 81·81-s − 41.1·83-s + 114·89-s − 187.·97-s − 193.·99-s − 134.·103-s + 198.·107-s + 74·109-s + ⋯ |
| L(s) = 1 | − 1.93·7-s + 9-s − 1.95·11-s + 25-s + 1.11·31-s + 0.461·37-s − 1.25·41-s + 1.64·43-s + 0.655·47-s + 2.74·49-s + 1.26·59-s − 1.96·61-s − 1.93·63-s + 73-s + 3.78·77-s + 81-s − 0.496·83-s + 1.28·89-s − 1.93·97-s − 1.95·99-s − 1.30·103-s + 1.85·107-s + 0.678·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.161098814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161098814\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 73 | \( 1 - 73T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 13.5T + 49T^{2} \) |
| 11 | \( 1 + 21.5T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 34.4T + 961T^{2} \) |
| 37 | \( 1 - 17.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 51.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 30.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 74.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 119.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 41.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 114T + 7.92e3T^{2} \) |
| 97 | \( 1 + 187.T + 9.40e3T^{2} \) |
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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
−9.789890554640533430040148924233, −8.895899591110373895140912948310, −7.81612047443054810792262695038, −7.07286773622937804247075583276, −6.32901781881287028690962117051, −5.38988877090379872280296981524, −4.35302307829853836310659334384, −3.19747933418583329138750396455, −2.49493776023656625137916708305, −0.61896823440201322121433479343,
0.61896823440201322121433479343, 2.49493776023656625137916708305, 3.19747933418583329138750396455, 4.35302307829853836310659334384, 5.38988877090379872280296981524, 6.32901781881287028690962117051, 7.07286773622937804247075583276, 7.81612047443054810792262695038, 8.895899591110373895140912948310, 9.789890554640533430040148924233
