| L(s) = 1 | + 3.93·2-s + 9.73·3-s + 7.51·4-s + 21.9·5-s + 38.3·6-s + 0.0666·7-s − 1.90·8-s + 67.8·9-s + 86.3·10-s − 20.8·11-s + 73.1·12-s − 57.6·13-s + 0.262·14-s + 213.·15-s − 67.6·16-s − 93.6·17-s + 267.·18-s + 7.63·19-s + 164.·20-s + 0.648·21-s − 82.1·22-s + 23·23-s − 18.5·24-s + 355.·25-s − 227.·26-s + 397.·27-s + 0.500·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 1.87·3-s + 0.939·4-s + 1.96·5-s + 2.60·6-s + 0.00359·7-s − 0.0841·8-s + 2.51·9-s + 2.73·10-s − 0.571·11-s + 1.76·12-s − 1.23·13-s + 0.00500·14-s + 3.67·15-s − 1.05·16-s − 1.33·17-s + 3.49·18-s + 0.0922·19-s + 1.84·20-s + 0.00674·21-s − 0.796·22-s + 0.208·23-s − 0.157·24-s + 2.84·25-s − 1.71·26-s + 2.83·27-s + 0.00337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.01726874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.01726874\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 3.93T + 8T^{2} \) |
| 3 | \( 1 - 9.73T + 27T^{2} \) |
| 5 | \( 1 - 21.9T + 125T^{2} \) |
| 7 | \( 1 - 0.0666T + 343T^{2} \) |
| 11 | \( 1 + 20.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.63T + 6.85e3T^{2} \) |
| 31 | \( 1 + 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 312.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 28.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 29.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 580.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 342.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 805.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 663.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 290.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−9.814616812561458162687645156847, −9.332877371215228074408575739372, −8.594952403943666736642958931508, −7.24402173035323491350519156206, −6.50031749622961014699109854817, −5.28462166180907810919337471071, −4.59957010075374757344531167230, −3.29002556720973833208310503235, −2.38551219820436019153670259498, −2.01994689025814481235318406658,
2.01994689025814481235318406658, 2.38551219820436019153670259498, 3.29002556720973833208310503235, 4.59957010075374757344531167230, 5.28462166180907810919337471071, 6.50031749622961014699109854817, 7.24402173035323491350519156206, 8.594952403943666736642958931508, 9.332877371215228074408575739372, 9.814616812561458162687645156847
