| L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s + 1.73·6-s − 3.46·7-s − 1.73·8-s + 9-s + 0.999·12-s − 1.73·13-s − 5.99·14-s − 5·16-s + 1.73·17-s + 1.73·18-s − 6.92·19-s − 3.46·21-s + 6·23-s − 1.73·24-s − 2.99·26-s + 27-s − 3.46·28-s + 1.73·29-s + 4·31-s − 5.19·32-s + 2.99·34-s + 0.999·36-s + 11·37-s − 11.9·38-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.707·6-s − 1.30·7-s − 0.612·8-s + 0.333·9-s + 0.288·12-s − 0.480·13-s − 1.60·14-s − 1.25·16-s + 0.420·17-s + 0.408·18-s − 1.58·19-s − 0.755·21-s + 1.25·23-s − 0.353·24-s − 0.588·26-s + 0.192·27-s − 0.654·28-s + 0.321·29-s + 0.718·31-s − 0.918·32-s + 0.514·34-s + 0.166·36-s + 1.80·37-s − 1.94·38-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
| \(L(1)\) |
\(\approx\) |
\(3.277104240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.277104240\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.56626177808390425503802566062, −6.77263142478702577736009670529, −6.31257080390574645679109099904, −5.69709533654698555277739253123, −4.71837506330001720979415272207, −4.25481102489424678687943795489, −3.45900852155314022953775734056, −2.83839711710534354804617616625, −2.30672533754142861420352165627, −0.67152987784046778169232541463,
0.67152987784046778169232541463, 2.30672533754142861420352165627, 2.83839711710534354804617616625, 3.45900852155314022953775734056, 4.25481102489424678687943795489, 4.71837506330001720979415272207, 5.69709533654698555277739253123, 6.31257080390574645679109099904, 6.77263142478702577736009670529, 7.56626177808390425503802566062
