| L(s) = 1 | − 2.70·2-s + 2.09·3-s + 5.33·4-s − 1.11·5-s − 5.67·6-s − 3.78·7-s − 9.02·8-s + 1.39·9-s + 3.02·10-s + 3.88·11-s + 11.1·12-s − 13-s + 10.2·14-s − 2.34·15-s + 13.7·16-s + 1.61·17-s − 3.76·18-s − 3.02·19-s − 5.95·20-s − 7.93·21-s − 10.5·22-s + 2.76·23-s − 18.9·24-s − 3.75·25-s + 2.70·26-s − 3.37·27-s − 20.2·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 1.20·3-s + 2.66·4-s − 0.499·5-s − 2.31·6-s − 1.43·7-s − 3.19·8-s + 0.463·9-s + 0.957·10-s + 1.17·11-s + 3.22·12-s − 0.277·13-s + 2.74·14-s − 0.604·15-s + 3.44·16-s + 0.390·17-s − 0.887·18-s − 0.694·19-s − 1.33·20-s − 1.73·21-s − 2.24·22-s + 0.577·23-s − 3.86·24-s − 0.750·25-s + 0.531·26-s − 0.649·27-s − 3.81·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7880413053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7880413053\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 6.44T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 + 7.00T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 - 9.14T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.507626891307652220650307554663, −8.909953446791652804301868053364, −8.193216147288800364058781052596, −7.55202770073356692950788546174, −6.66889316514424798431877044159, −6.14480909909529535958433213371, −3.93469706627793419990498592482, −3.06115218507083121605705208913, −2.29857435495431403684391209584, −0.791521893186328783452432589949,
0.791521893186328783452432589949, 2.29857435495431403684391209584, 3.06115218507083121605705208913, 3.93469706627793419990498592482, 6.14480909909529535958433213371, 6.66889316514424798431877044159, 7.55202770073356692950788546174, 8.193216147288800364058781052596, 8.909953446791652804301868053364, 9.507626891307652220650307554663
