| L(s) = 1 | + 0.780·2-s − 3·3-s − 7.39·4-s − 21.9·5-s − 2.34·6-s + 32.1·7-s − 12.0·8-s + 9·9-s − 17.0·10-s − 35.3·11-s + 22.1·12-s + 34.5·13-s + 25.0·14-s + 65.7·15-s + 49.7·16-s − 27.9·17-s + 7.02·18-s + 74.3·19-s + 162.·20-s − 96.3·21-s − 27.5·22-s + 128.·23-s + 36.0·24-s + 355.·25-s + 26.9·26-s − 27·27-s − 237.·28-s + ⋯ |
| L(s) = 1 | + 0.275·2-s − 0.577·3-s − 0.923·4-s − 1.96·5-s − 0.159·6-s + 1.73·7-s − 0.530·8-s + 0.333·9-s − 0.540·10-s − 0.967·11-s + 0.533·12-s + 0.736·13-s + 0.478·14-s + 1.13·15-s + 0.777·16-s − 0.399·17-s + 0.0919·18-s + 0.897·19-s + 1.81·20-s − 1.00·21-s − 0.266·22-s + 1.16·23-s + 0.306·24-s + 2.84·25-s + 0.203·26-s − 0.192·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9191737869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9191737869\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 - 0.780T + 8T^{2} \) |
| 5 | \( 1 + 21.9T + 125T^{2} \) |
| 7 | \( 1 - 32.1T + 343T^{2} \) |
| 11 | \( 1 + 35.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 377.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 261.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 113.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 337.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 168.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 805.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 797.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 251.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 653.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.57e3T + 9.12e5T^{2} \) |
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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
−12.11824608556285494997266444973, −11.25602678153488652485473977365, −10.77831674649618362174415066783, −8.848434346641670044491190720846, −8.080171826592934420480470558090, −7.33806983536757041267130994978, −5.24818082800834043213733079606, −4.67173796312431542786644436872, −3.57837690729454845491835046412, −0.76509155334516902874932818328,
0.76509155334516902874932818328, 3.57837690729454845491835046412, 4.67173796312431542786644436872, 5.24818082800834043213733079606, 7.33806983536757041267130994978, 8.080171826592934420480470558090, 8.848434346641670044491190720846, 10.77831674649618362174415066783, 11.25602678153488652485473977365, 12.11824608556285494997266444973
