| L(s) = 1 | + 2-s + 3.15·3-s + 4-s − 3.25·5-s + 3.15·6-s − 2.03·7-s + 8-s + 6.96·9-s − 3.25·10-s + 5.04·11-s + 3.15·12-s + 6.29·13-s − 2.03·14-s − 10.2·15-s + 16-s − 7.12·17-s + 6.96·18-s − 0.944·19-s − 3.25·20-s − 6.42·21-s + 5.04·22-s + 2.93·23-s + 3.15·24-s + 5.57·25-s + 6.29·26-s + 12.5·27-s − 2.03·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.82·3-s + 0.5·4-s − 1.45·5-s + 1.28·6-s − 0.769·7-s + 0.353·8-s + 2.32·9-s − 1.02·10-s + 1.52·11-s + 0.911·12-s + 1.74·13-s − 0.544·14-s − 2.65·15-s + 0.250·16-s − 1.72·17-s + 1.64·18-s − 0.216·19-s − 0.727·20-s − 1.40·21-s + 1.07·22-s + 0.611·23-s + 0.644·24-s + 1.11·25-s + 1.23·26-s + 2.40·27-s − 0.384·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.373953524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.373953524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 0.944T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 + 0.237T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 79 | \( 1 - 0.114T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
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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
−8.260211912061513257712250995684, −7.32142881728434997938039032589, −6.68477048465220447998455766123, −6.36807882974613184424752906011, −4.69526667558193097083485968901, −3.96880509006492783467584513940, −3.73716977127052721859492601819, −3.16560752173945468293894895010, −2.16778833643551915382203918349, −1.06533208583903506453319760092,
1.06533208583903506453319760092, 2.16778833643551915382203918349, 3.16560752173945468293894895010, 3.73716977127052721859492601819, 3.96880509006492783467584513940, 4.69526667558193097083485968901, 6.36807882974613184424752906011, 6.68477048465220447998455766123, 7.32142881728434997938039032589, 8.260211912061513257712250995684
