| 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
−8.194852569258082306895102460517, −7.38085032783333951021334054698, −6.85851436603056453135442244727, −6.09298564609103753549502669358, −4.82752085943032959478442046847, −4.30867533687546701303580636894, −3.47581971954528249109909999818, −2.55274790591855405682823675014, −0.925805875576685172252377035531, −0.37526258179975177899230333491,
1.49344973670192789331363840640, 2.42541325629103031811563914766, 3.23117812675103618019252359158, 4.41608674006652361791320444835, 5.08321725534977003477079580794, 5.93309734457513798116128043172, 6.62026587064162373683977965788, 7.57058660091882332050432995037, 8.335917560580566895414080046721, 8.830853172723975910678557142697
