| L(s) = 1 | + 2.47·2-s + 3.33·3-s + 4.12·4-s − 1.85·5-s + 8.26·6-s − 1.00·7-s + 5.26·8-s + 8.14·9-s − 4.59·10-s − 1.67·11-s + 13.7·12-s − 13-s − 2.47·14-s − 6.19·15-s + 4.78·16-s − 2.81·17-s + 20.1·18-s − 4.29·19-s − 7.65·20-s − 3.34·21-s − 4.14·22-s + 6.60·23-s + 17.5·24-s − 1.56·25-s − 2.47·26-s + 17.1·27-s − 4.13·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 1.92·3-s + 2.06·4-s − 0.829·5-s + 3.37·6-s − 0.378·7-s + 1.86·8-s + 2.71·9-s − 1.45·10-s − 0.504·11-s + 3.97·12-s − 0.277·13-s − 0.662·14-s − 1.59·15-s + 1.19·16-s − 0.683·17-s + 4.75·18-s − 0.985·19-s − 1.71·20-s − 0.729·21-s − 0.882·22-s + 1.37·23-s + 3.58·24-s − 0.312·25-s − 0.485·26-s + 3.30·27-s − 0.780·28-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
| \(L(1)\) |
\(\approx\) |
\(7.182054072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.182054072\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 3.33T + 3T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 - 0.333T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 - 1.96T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 6.00T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 - 6.83T + 61T^{2} \) |
| 67 | \( 1 - 0.870T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 6.06T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−9.557734193534840857303857836912, −8.545359672997063223000453243100, −7.944748832937396043342268490101, −7.03534419269921728987281296127, −6.49345444534890812848156237972, −4.86559568378688786997447979880, −4.35114715778466146919256614904, −3.43456964973660516575925810137, −2.90957985322583328367711257456, −2.00962762284363084843109439967,
2.00962762284363084843109439967, 2.90957985322583328367711257456, 3.43456964973660516575925810137, 4.35114715778466146919256614904, 4.86559568378688786997447979880, 6.49345444534890812848156237972, 7.03534419269921728987281296127, 7.944748832937396043342268490101, 8.545359672997063223000453243100, 9.557734193534840857303857836912
