| L(s) = 1 | − 2.23·2-s + 2.99·4-s − 2.10·5-s − 1.54·7-s − 2.22·8-s + 4.70·10-s + 11-s − 7.11·13-s + 3.46·14-s − 1.01·16-s − 3.21·17-s − 3.37·19-s − 6.31·20-s − 2.23·22-s + 5.08·23-s − 0.562·25-s + 15.9·26-s − 4.64·28-s + 2.29·29-s − 2.78·31-s + 6.71·32-s + 7.18·34-s + 3.26·35-s − 9.34·37-s + 7.54·38-s + 4.69·40-s − 0.774·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.49·4-s − 0.942·5-s − 0.585·7-s − 0.788·8-s + 1.48·10-s + 0.301·11-s − 1.97·13-s + 0.925·14-s − 0.252·16-s − 0.780·17-s − 0.773·19-s − 1.41·20-s − 0.476·22-s + 1.06·23-s − 0.112·25-s + 3.12·26-s − 0.877·28-s + 0.427·29-s − 0.500·31-s + 1.18·32-s + 1.23·34-s + 0.551·35-s − 1.53·37-s + 1.22·38-s + 0.742·40-s − 0.121·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03322289774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03322289774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 13 | \( 1 + 7.11T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 5.08T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 + 0.774T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 + 9.43T + 47T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 + 8.69T + 59T^{2} \) |
| 67 | \( 1 - 0.876T + 67T^{2} \) |
| 71 | \( 1 + 8.01T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.110333817112859379807766355259, −7.54575350659268404650264052744, −6.80297601936500735442741868365, −6.58952227791135569926384922840, −5.08681935078812889360066683904, −4.49798815470461786092833317933, −3.41330835157544665099429882832, −2.53730446231359490987162542305, −1.63156956823004169871663155259, −0.11803060236078396550123144541,
0.11803060236078396550123144541, 1.63156956823004169871663155259, 2.53730446231359490987162542305, 3.41330835157544665099429882832, 4.49798815470461786092833317933, 5.08681935078812889360066683904, 6.58952227791135569926384922840, 6.80297601936500735442741868365, 7.54575350659268404650264052744, 8.110333817112859379807766355259
