| L(s) = 1 | − 13.3·5-s − 15.3·7-s − 24.9·11-s + 13·13-s − 65.5·17-s + 73.1·19-s − 28.5·23-s + 54.0·25-s − 220.·29-s − 138.·31-s + 205.·35-s − 354.·37-s − 297.·41-s + 9.62·43-s + 219.·47-s − 106.·49-s − 189.·53-s + 333.·55-s − 329.·59-s − 838.·61-s − 173.·65-s + 386.·67-s − 664.·71-s + 248.·73-s + 383.·77-s + 1.26e3·79-s + 157.·83-s + ⋯ |
| L(s) = 1 | − 1.19·5-s − 0.830·7-s − 0.682·11-s + 0.277·13-s − 0.934·17-s + 0.883·19-s − 0.259·23-s + 0.432·25-s − 1.41·29-s − 0.805·31-s + 0.993·35-s − 1.57·37-s − 1.13·41-s + 0.0341·43-s + 0.681·47-s − 0.310·49-s − 0.490·53-s + 0.816·55-s − 0.726·59-s − 1.76·61-s − 0.331·65-s + 0.705·67-s − 1.11·71-s + 0.398·73-s + 0.566·77-s + 1.80·79-s + 0.208·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3809326947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3809326947\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 13.3T + 125T^{2} \) |
| 7 | \( 1 + 15.3T + 343T^{2} \) |
| 11 | \( 1 + 24.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 354.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 9.62T + 7.95e4T^{2} \) |
| 47 | \( 1 - 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 189.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 329.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 838.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 386.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 664.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 248.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 157.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.902921733716050199424111499459, −7.958628741693447049171440367269, −7.39157479292450407604174587854, −6.63320909834483675107415026896, −5.63862862452443894890860570511, −4.74149378628280901363420889326, −3.68626123522913615522156940459, −3.22418992306099529151945928712, −1.88231142312022988005247909335, −0.27445774622771076669377282560,
0.27445774622771076669377282560, 1.88231142312022988005247909335, 3.22418992306099529151945928712, 3.68626123522913615522156940459, 4.74149378628280901363420889326, 5.63862862452443894890860570511, 6.63320909834483675107415026896, 7.39157479292450407604174587854, 7.958628741693447049171440367269, 8.902921733716050199424111499459
