| L(s) = 1 | + 1.82·2-s − 3-s + 1.33·4-s − 0.563·5-s − 1.82·6-s + 3.34·7-s − 1.21·8-s + 9-s − 1.02·10-s + 6.11·11-s − 1.33·12-s + 2.34·13-s + 6.11·14-s + 0.563·15-s − 4.88·16-s + 6.16·17-s + 1.82·18-s − 4.83·19-s − 0.753·20-s − 3.34·21-s + 11.1·22-s − 2.50·23-s + 1.21·24-s − 4.68·25-s + 4.28·26-s − 27-s + 4.47·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.668·4-s − 0.251·5-s − 0.745·6-s + 1.26·7-s − 0.428·8-s + 0.333·9-s − 0.325·10-s + 1.84·11-s − 0.385·12-s + 0.650·13-s + 1.63·14-s + 0.145·15-s − 1.22·16-s + 1.49·17-s + 0.430·18-s − 1.11·19-s − 0.168·20-s − 0.730·21-s + 2.38·22-s − 0.521·23-s + 0.247·24-s − 0.936·25-s + 0.840·26-s − 0.192·27-s + 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330239462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330239462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 5 | \( 1 + 0.563T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 - 0.750T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 + 7.06T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−11.54978570516836193553446127277, −11.02098163011380485923199251662, −9.553961051066255415725666084479, −8.516296022239717123669414588990, −7.34409498701367195557324982586, −6.13316256483590013361967559290, −5.51783641871794438086446835130, −4.24171092207917730202305869727, −3.78151993291473791399775376071, −1.64157834670226032765876295070,
1.64157834670226032765876295070, 3.78151993291473791399775376071, 4.24171092207917730202305869727, 5.51783641871794438086446835130, 6.13316256483590013361967559290, 7.34409498701367195557324982586, 8.516296022239717123669414588990, 9.553961051066255415725666084479, 11.02098163011380485923199251662, 11.54978570516836193553446127277
