| L(s) = 1 | + 2.26·2-s − 1.74·3-s + 3.13·4-s − 5-s − 3.96·6-s + 4.36·7-s + 2.56·8-s + 0.0583·9-s − 2.26·10-s + 3.95·11-s − 5.47·12-s − 0.563·13-s + 9.89·14-s + 1.74·15-s − 0.458·16-s − 17-s + 0.132·18-s − 1.24·19-s − 3.13·20-s − 7.64·21-s + 8.96·22-s + 2.58·23-s − 4.48·24-s + 25-s − 1.27·26-s + 5.14·27-s + 13.6·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 1.00·3-s + 1.56·4-s − 0.447·5-s − 1.61·6-s + 1.65·7-s + 0.905·8-s + 0.0194·9-s − 0.716·10-s + 1.19·11-s − 1.58·12-s − 0.156·13-s + 2.64·14-s + 0.451·15-s − 0.114·16-s − 0.242·17-s + 0.0311·18-s − 0.286·19-s − 0.700·20-s − 1.66·21-s + 1.91·22-s + 0.538·23-s − 0.914·24-s + 0.200·25-s − 0.250·26-s + 0.990·27-s + 2.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.190326571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.190326571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 0.563T + 13T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 1.16T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 - 5.75T + 41T^{2} \) |
| 43 | \( 1 + 5.77T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 73 | \( 1 - 8.51T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 2.46T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
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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
−7.930700527853992832602630089372, −6.89315383480837509329169506231, −6.53999521247170891372751017583, −5.66283428488066755939678725714, −5.12208863268123551922045622971, −4.49777745399947133627285388870, −4.09729464961412990885563557251, −3.02674817665193009473319029562, −1.99774565537830479844347534520, −0.925336826087079717834416233375,
0.925336826087079717834416233375, 1.99774565537830479844347534520, 3.02674817665193009473319029562, 4.09729464961412990885563557251, 4.49777745399947133627285388870, 5.12208863268123551922045622971, 5.66283428488066755939678725714, 6.53999521247170891372751017583, 6.89315383480837509329169506231, 7.930700527853992832602630089372
