| L(s) = 1 | − 1.43·2-s + 2.33·3-s + 0.0606·4-s + 2.88·5-s − 3.34·6-s + 3.63·7-s + 2.78·8-s + 2.44·9-s − 4.14·10-s + 3.33·11-s + 0.141·12-s − 13-s − 5.22·14-s + 6.73·15-s − 4.11·16-s − 2.35·17-s − 3.50·18-s + 3.81·19-s + 0.175·20-s + 8.49·21-s − 4.78·22-s + 7.05·23-s + 6.49·24-s + 3.33·25-s + 1.43·26-s − 1.29·27-s + 0.220·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 1.34·3-s + 0.0303·4-s + 1.29·5-s − 1.36·6-s + 1.37·7-s + 0.984·8-s + 0.814·9-s − 1.31·10-s + 1.00·11-s + 0.0408·12-s − 0.277·13-s − 1.39·14-s + 1.73·15-s − 1.02·16-s − 0.572·17-s − 0.827·18-s + 0.874·19-s + 0.0391·20-s + 1.85·21-s − 1.01·22-s + 1.47·23-s + 1.32·24-s + 0.667·25-s + 0.281·26-s − 0.249·27-s + 0.0417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.218373376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218373376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 7.40T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 4.31T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
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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
−9.337595717903021400542967734836, −9.039362496836275414698116967477, −8.183270864893105294596193295783, −7.59988938587430515282674132643, −6.66210625390727669695700924889, −5.29686361502187680934614434265, −4.53738911808375691277580656179, −3.23023256971380008687334596063, −1.86711385717447295887905884257, −1.52323386256644983499671358962,
1.52323386256644983499671358962, 1.86711385717447295887905884257, 3.23023256971380008687334596063, 4.53738911808375691277580656179, 5.29686361502187680934614434265, 6.66210625390727669695700924889, 7.59988938587430515282674132643, 8.183270864893105294596193295783, 9.039362496836275414698116967477, 9.337595717903021400542967734836
