| L(s) = 1 | + 2.41·2-s + 2.41·3-s + 3.82·4-s + 2·5-s + 5.82·6-s − 3.82·7-s + 4.41·8-s + 2.82·9-s + 4.82·10-s + 9.24·12-s + 5·13-s − 9.24·14-s + 4.82·15-s + 2.99·16-s + 4.41·17-s + 6.82·18-s − 5.24·19-s + 7.65·20-s − 9.24·21-s + 3.17·23-s + 10.6·24-s − 25-s + 12.0·26-s − 0.414·27-s − 14.6·28-s + 29-s + 11.6·30-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.39·3-s + 1.91·4-s + 0.894·5-s + 2.37·6-s − 1.44·7-s + 1.56·8-s + 0.942·9-s + 1.52·10-s + 2.66·12-s + 1.38·13-s − 2.47·14-s + 1.24·15-s + 0.749·16-s + 1.07·17-s + 1.60·18-s − 1.20·19-s + 1.71·20-s − 2.01·21-s + 0.661·23-s + 2.17·24-s − 0.200·25-s + 2.36·26-s − 0.0797·27-s − 2.76·28-s + 0.185·29-s + 2.12·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.323961005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.323961005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 9.24T + 97T^{2} \) |
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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
−8.511468521136550993252927669522, −7.51483316664061613205271064009, −6.64675502714306942259424180344, −6.04118977497786049401995843092, −5.68215834351972005705294834114, −4.31514482585859362862021316585, −3.78977069372423707882443427392, −2.92332821871196718639064209232, −2.68979780369886358348184156585, −1.50375815087984065269763681125,
1.50375815087984065269763681125, 2.68979780369886358348184156585, 2.92332821871196718639064209232, 3.78977069372423707882443427392, 4.31514482585859362862021316585, 5.68215834351972005705294834114, 6.04118977497786049401995843092, 6.64675502714306942259424180344, 7.51483316664061613205271064009, 8.511468521136550993252927669522
