L(s) = 1 | + (3.32 + 3.32i)5-s − 4.04·7-s + (6.82 − 6.82i)11-s + (−4.29 + 4.29i)13-s − 30.1·17-s + (19.7 + 19.7i)19-s + 28.2·23-s − 2.86i·25-s + (−21.3 + 21.3i)29-s + 38.0i·31-s + (−13.4 − 13.4i)35-s + (42.8 + 42.8i)37-s − 48.2i·41-s + (−32.6 + 32.6i)43-s + 15.8i·47-s + ⋯ |
L(s) = 1 | + (0.665 + 0.665i)5-s − 0.577·7-s + (0.620 − 0.620i)11-s + (−0.330 + 0.330i)13-s − 1.77·17-s + (1.03 + 1.03i)19-s + 1.22·23-s − 0.114i·25-s + (−0.736 + 0.736i)29-s + 1.22i·31-s + (−0.384 − 0.384i)35-s + (1.15 + 1.15i)37-s − 1.17i·41-s + (−0.759 + 0.759i)43-s + 0.336i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.483748545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483748545\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.32 - 3.32i)T + 25iT^{2} \) |
| 7 | \( 1 + 4.04T + 49T^{2} \) |
| 11 | \( 1 + (-6.82 + 6.82i)T - 121iT^{2} \) |
| 13 | \( 1 + (4.29 - 4.29i)T - 169iT^{2} \) |
| 17 | \( 1 + 30.1T + 289T^{2} \) |
| 19 | \( 1 + (-19.7 - 19.7i)T + 361iT^{2} \) |
| 23 | \( 1 - 28.2T + 529T^{2} \) |
| 29 | \( 1 + (21.3 - 21.3i)T - 841iT^{2} \) |
| 31 | \( 1 - 38.0iT - 961T^{2} \) |
| 37 | \( 1 + (-42.8 - 42.8i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (32.6 - 32.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 15.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (0.476 + 0.476i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-9.97 + 9.97i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (37.9 - 37.9i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (20.0 + 20.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 40.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 30.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 130. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (2.26 + 2.26i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 72.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112.T + 9.40e3T^{2} \) |
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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
−9.737564620311732457373217209594, −9.200572482950069009079143672454, −8.348881872353473826349628494496, −7.03713489524879649675310678921, −6.62464985009653395625260025684, −5.79566211585722488552790123969, −4.71357108469345924566544221576, −3.49926500422904746976671494515, −2.68364952263024548611170505726, −1.39375296383116913231068436262,
0.44142911537820883525086112630, 1.85284537543137185336629613286, 2.91259413885066386545139979760, 4.27276761688209386486342514352, 5.00932719170165790990987679917, 5.99260854355606202527088716291, 6.84465665263159403409005984506, 7.58459126854219687185336689228, 8.906627843706521713873654588682, 9.338657796654066787212245196170