L(s) = 1 | + 2.82·2-s − 1.75·3-s + 8.00·4-s − 11.1i·5-s − 4.96·6-s + 30.3i·7-s + 22.6·8-s − 77.9·9-s − 31.6i·10-s − 137. i·11-s − 14.0·12-s + 219.·13-s + 85.8i·14-s + 19.6i·15-s + 64.0·16-s − 347. i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.195·3-s + 0.500·4-s − 0.447i·5-s − 0.137·6-s + 0.619i·7-s + 0.353·8-s − 0.961·9-s − 0.316i·10-s − 1.13i·11-s − 0.0975·12-s + 1.29·13-s + 0.437i·14-s + 0.0872i·15-s + 0.250·16-s − 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.401205113\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.401205113\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82T \) |
| 5 | \( 1 + 11.1iT \) |
| 23 | \( 1 + (-513. + 126. i)T \) |
good | 3 | \( 1 + 1.75T + 81T^{2} \) |
| 7 | \( 1 - 30.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 137. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 219.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 347. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 218. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 268.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 37.9T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.01e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.35e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.60e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.89e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.74e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.39e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.93e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.83e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.62e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.59e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.13e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.39e3iT - 8.85e7T^{2} \) |
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
−11.30603745725336505464431492557, −10.93322230954067660358631094525, −9.034031128282790943592855822002, −8.637078493761821306821974103484, −7.10325868183357130682656686428, −5.75878228526554120649430907577, −5.37436385227034113229755812177, −3.73615676442136742073304182189, −2.58772741975536837377020062247, −0.67008232810627061943385336033,
1.56395287408167712097315067999, 3.20656789695508384562797652588, 4.22590744449223004370977116777, 5.61461860121062341473843584816, 6.49641962292995124196313647573, 7.55271086450448432286510092008, 8.708513350832558897080519469828, 10.16031226700806107769998546108, 10.93104472464201854434365909455, 11.68412570663371878043317191588