L(s) = 1 | − 2.23·5-s + 12.3i·7-s + 11.0i·11-s − 2.82·13-s − 6.52·17-s + 27.9i·19-s + 7.90i·23-s + 5.00·25-s + 50.7·29-s + 36.3i·31-s − 27.7i·35-s + 18.9·37-s − 5.30·41-s + 45.5i·43-s − 11.7i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.77i·7-s + 1.00i·11-s − 0.216·13-s − 0.383·17-s + 1.47i·19-s + 0.343i·23-s + 0.200·25-s + 1.74·29-s + 1.17i·31-s − 0.791i·35-s + 0.511·37-s − 0.129·41-s + 1.06i·43-s − 0.249i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.337968865\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.337968865\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
good | 7 | \( 1 - 12.3iT - 49T^{2} \) |
| 11 | \( 1 - 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 2.82T + 169T^{2} \) |
| 17 | \( 1 + 6.52T + 289T^{2} \) |
| 19 | \( 1 - 27.9iT - 361T^{2} \) |
| 23 | \( 1 - 7.90iT - 529T^{2} \) |
| 29 | \( 1 - 50.7T + 841T^{2} \) |
| 31 | \( 1 - 36.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 45.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 99.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 127.T + 9.40e3T^{2} \) |
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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
−8.830853172723975910678557142697, −8.335917560580566895414080046721, −7.57058660091882332050432995037, −6.62026587064162373683977965788, −5.93309734457513798116128043172, −5.08321725534977003477079580794, −4.41608674006652361791320444835, −3.23117812675103618019252359158, −2.42541325629103031811563914766, −1.49344973670192789331363840640,
0.37526258179975177899230333491, 0.925805875576685172252377035531, 2.55274790591855405682823675014, 3.47581971954528249109909999818, 4.30867533687546701303580636894, 4.82752085943032959478442046847, 6.09298564609103753549502669358, 6.85851436603056453135442244727, 7.38085032783333951021334054698, 8.194852569258082306895102460517