| L(s) = 1 | + 3.85·2-s − 3·3-s + 6.86·4-s − 3.72·5-s − 11.5·6-s − 16.4·7-s − 4.38·8-s + 9·9-s − 14.3·10-s − 11·11-s − 20.5·12-s − 48.3·13-s − 63.4·14-s + 11.1·15-s − 71.8·16-s + 50.7·17-s + 34.6·18-s − 92.8·19-s − 25.5·20-s + 49.3·21-s − 42.4·22-s − 29.7·23-s + 13.1·24-s − 111.·25-s − 186.·26-s − 27·27-s − 112.·28-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 0.577·3-s + 0.857·4-s − 0.333·5-s − 0.786·6-s − 0.888·7-s − 0.193·8-s + 0.333·9-s − 0.454·10-s − 0.301·11-s − 0.495·12-s − 1.03·13-s − 1.21·14-s + 0.192·15-s − 1.12·16-s + 0.723·17-s + 0.454·18-s − 1.12·19-s − 0.286·20-s + 0.513·21-s − 0.410·22-s − 0.269·23-s + 0.111·24-s − 0.888·25-s − 1.40·26-s − 0.192·27-s − 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.571789550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571789550\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.85T + 8T^{2} \) |
| 5 | \( 1 + 3.72T + 125T^{2} \) |
| 7 | \( 1 + 16.4T + 343T^{2} \) |
| 13 | \( 1 + 48.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 29.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 176.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 716.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 852.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 766.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 928.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 499.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 176.T + 9.12e5T^{2} \) |
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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
−8.815829385330614742334890561422, −7.70565319490952113290826896123, −6.97237895255649264117279586039, −6.06347345692178006498684537381, −5.68815747042985370973525395798, −4.60209755771164080207553833484, −4.11124541165312394839030938137, −3.09845385972599954224257810170, −2.27346707931281746462447548333, −0.45224615132438645843837464673,
0.45224615132438645843837464673, 2.27346707931281746462447548333, 3.09845385972599954224257810170, 4.11124541165312394839030938137, 4.60209755771164080207553833484, 5.68815747042985370973525395798, 6.06347345692178006498684537381, 6.97237895255649264117279586039, 7.70565319490952113290826896123, 8.815829385330614742334890561422
