L(s) = 1 | + (−1.32 + 0.5i)2-s + (1.50 − 1.32i)4-s − i·5-s + 7-s + (−1.32 + 2.50i)8-s + (0.5 + 1.32i)10-s − 3i·11-s − 5.29i·13-s + (−1.32 + 0.5i)14-s + (0.500 − 3.96i)16-s + 5.29·17-s + 5.29i·19-s + (−1.32 − 1.50i)20-s + (1.5 + 3.96i)22-s + 5.29·23-s + ⋯ |
L(s) = 1 | + (−0.935 + 0.353i)2-s + (0.750 − 0.661i)4-s − 0.447i·5-s + 0.377·7-s + (−0.467 + 0.883i)8-s + (0.158 + 0.418i)10-s − 0.904i·11-s − 1.46i·13-s + (−0.353 + 0.133i)14-s + (0.125 − 0.992i)16-s + 1.28·17-s + 1.21i·19-s + (−0.295 − 0.335i)20-s + (0.319 + 0.846i)22-s + 1.10·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.816616 - 0.202739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.816616 - 0.202739i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.32 - 0.5i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 5.29iT - 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 5.29iT - 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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
−12.10834075075275532009648511154, −10.99766688994302574508339754702, −10.27351880333569222077791642321, −9.183393638590664607059590793500, −8.166995065736436581086886298015, −7.61564852060630932565362852007, −6.00240492982699520370941990099, −5.26727410910163239632552261105, −3.15764758579145291204500908366, −1.09597606645478970806256319479,
1.75430672134053610056750135567, 3.27823863425210543679099280084, 4.91339111600050188067292608053, 6.83491221339109599861053635000, 7.23618476599573020784224113199, 8.671047688937859713956696401459, 9.427389652778808240260082855744, 10.45687289658124994464006100684, 11.30793832299904834146812729396, 12.08665782771352432742843186882