| L(s) = 1 | − 2.76·2-s + 3-s + 5.62·4-s − 2.76·6-s + 1.86·7-s − 10.0·8-s + 9-s + 5.62·12-s + 4.62·13-s − 5.14·14-s + 16.4·16-s + 2.49·17-s − 2.76·18-s + 5.38·19-s + 1.86·21-s − 7.14·23-s − 10.0·24-s − 12.7·26-s + 27-s + 10.4·28-s + 3.52·29-s + 8.62·31-s − 25.2·32-s − 6.87·34-s + 5.62·36-s + 8.87·37-s − 14.8·38-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.81·4-s − 1.12·6-s + 0.704·7-s − 3.54·8-s + 0.333·9-s + 1.62·12-s + 1.28·13-s − 1.37·14-s + 4.10·16-s + 0.604·17-s − 0.650·18-s + 1.23·19-s + 0.406·21-s − 1.49·23-s − 2.04·24-s − 2.50·26-s + 0.192·27-s + 1.98·28-s + 0.654·29-s + 1.54·31-s − 4.46·32-s − 1.17·34-s + 0.937·36-s + 1.45·37-s − 2.41·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481887782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481887782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 - 0.761T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 0.373T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 + 4.42T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.77T + 73T^{2} \) |
| 79 | \( 1 - 6.01T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 9.04T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.950572161123349481480619899591, −7.55581255724791506258141289812, −6.49614565028758058902556663634, −6.16177331898411215237106747813, −5.15445332491733764757124599151, −3.89757509607740035571604150677, −3.09688086523261067651874353973, −2.31536321485713055993466343543, −1.41376748294642212951909696495, −0.863922461057985756973145106078,
0.863922461057985756973145106078, 1.41376748294642212951909696495, 2.31536321485713055993466343543, 3.09688086523261067651874353973, 3.89757509607740035571604150677, 5.15445332491733764757124599151, 6.16177331898411215237106747813, 6.49614565028758058902556663634, 7.55581255724791506258141289812, 7.950572161123349481480619899591
