| L(s) = 1 | + 1.11·2-s + 0.352·3-s − 6.75·4-s + 4.08·5-s + 0.392·6-s − 16.4·8-s − 26.8·9-s + 4.55·10-s − 52.0·11-s − 2.38·12-s − 36.6·13-s + 1.44·15-s + 35.7·16-s − 124.·17-s − 29.9·18-s + 32.5·19-s − 27.6·20-s − 58.0·22-s + 68.0·23-s − 5.79·24-s − 108.·25-s − 40.8·26-s − 18.9·27-s + 65.7·29-s + 1.60·30-s − 305.·31-s + 171.·32-s + ⋯ |
| L(s) = 1 | + 0.393·2-s + 0.0677·3-s − 0.844·4-s + 0.365·5-s + 0.0267·6-s − 0.726·8-s − 0.995·9-s + 0.144·10-s − 1.42·11-s − 0.0572·12-s − 0.781·13-s + 0.0247·15-s + 0.558·16-s − 1.77·17-s − 0.392·18-s + 0.393·19-s − 0.309·20-s − 0.562·22-s + 0.616·23-s − 0.0492·24-s − 0.866·25-s − 0.307·26-s − 0.135·27-s + 0.420·29-s + 0.00976·30-s − 1.76·31-s + 0.946·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2947064634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2947064634\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 1.11T + 8T^{2} \) |
| 3 | \( 1 - 0.352T + 27T^{2} \) |
| 5 | \( 1 - 4.08T + 125T^{2} \) |
| 11 | \( 1 + 52.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 65.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 131.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 573.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 233.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 511.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 175.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 742.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.888770338261300329175692148398, −7.958024451258146151666702557751, −7.14463921863355448985576183054, −6.07280710067556265895087349652, −5.24558244977141776942086576411, −4.96057335606314211158726554688, −3.78588953817961089144354337010, −2.84620451416305998752396593837, −2.07200023496538190942596448724, −0.21170216108802380497389006006,
0.21170216108802380497389006006, 2.07200023496538190942596448724, 2.84620451416305998752396593837, 3.78588953817961089144354337010, 4.96057335606314211158726554688, 5.24558244977141776942086576411, 6.07280710067556265895087349652, 7.14463921863355448985576183054, 7.958024451258146151666702557751, 8.888770338261300329175692148398
