| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 2.59i·5-s − 3.01·7-s − 2.82i·8-s + 3.67·10-s + 4.26i·11-s + 7.88·13-s − 4.26i·14-s + 4.00·16-s + 10.6i·17-s + 36.1·19-s + 5.19i·20-s − 6.02·22-s + 4.79i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.519i·5-s − 0.430·7-s − 0.353i·8-s + 0.367·10-s + 0.387i·11-s + 0.606·13-s − 0.304i·14-s + 0.250·16-s + 0.625i·17-s + 1.90·19-s + 0.259i·20-s − 0.273·22-s + 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41994 + 0.735019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41994 + 0.735019i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 5 | \( 1 + 2.59iT - 25T^{2} \) |
| 7 | \( 1 + 3.01T + 49T^{2} \) |
| 11 | \( 1 - 4.26iT - 121T^{2} \) |
| 13 | \( 1 - 7.88T + 169T^{2} \) |
| 17 | \( 1 - 10.6iT - 289T^{2} \) |
| 19 | \( 1 - 36.1T + 361T^{2} \) |
| 29 | \( 1 - 4.54iT - 841T^{2} \) |
| 31 | \( 1 - 33.1T + 961T^{2} \) |
| 37 | \( 1 - 21.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 2.40iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 49.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.23T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 41.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 90.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.6T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14383083154701096478372146808, −9.950912468130746149502400829634, −9.275323021561029459704256279948, −8.321180075616087445862827209445, −7.44110717340327439424272899911, −6.40899717773865913924659004007, −5.47036568509685482657784877311, −4.44383292268145472564960387900, −3.17679860451230248840237172098, −1.11059453999483769836980345811,
0.929219801866653148123366992566, 2.74134282628985184801358454528, 3.51640308257900153115395543463, 4.91789375077163611221705409408, 6.06647957135124697659261801204, 7.14165594856331117269254854272, 8.231811440172670993531090474142, 9.314337138898502863284943913201, 9.991715289874110214793705588832, 10.97726400024580184451502973893
