L(s) = 1 | + (−1.78 − 0.741i)5-s + (2.33 − 2.33i)7-s + (−0.683 − 0.283i)11-s + (2.43 + 5.88i)13-s − 0.967·17-s + (2.52 − 1.04i)19-s + (5.00 − 5.00i)23-s + (−0.882 − 0.882i)25-s + (0.563 + 1.36i)29-s − 7.28i·31-s + (−5.91 + 2.45i)35-s + (3.26 − 7.88i)37-s + (−6.80 − 6.80i)41-s + (−1.36 + 3.29i)43-s − 3.69i·47-s + ⋯ |
L(s) = 1 | + (−0.800 − 0.331i)5-s + (0.883 − 0.883i)7-s + (−0.206 − 0.0853i)11-s + (0.675 + 1.63i)13-s − 0.234·17-s + (0.579 − 0.240i)19-s + (1.04 − 1.04i)23-s + (−0.176 − 0.176i)25-s + (0.104 + 0.252i)29-s − 1.30i·31-s + (−0.999 + 0.414i)35-s + (0.536 − 1.29i)37-s + (−1.06 − 1.06i)41-s + (−0.208 + 0.502i)43-s − 0.538i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.488303131\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.488303131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.78 + 0.741i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 2.33i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.683 + 0.283i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.43 - 5.88i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 0.967T + 17T^{2} \) |
| 19 | \( 1 + (-2.52 + 1.04i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.00 + 5.00i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.563 - 1.36i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 7.28iT - 31T^{2} \) |
| 37 | \( 1 + (-3.26 + 7.88i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (6.80 + 6.80i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.36 - 3.29i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 3.69iT - 47T^{2} \) |
| 53 | \( 1 + (1.85 - 4.47i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.05 + 9.79i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 4.48i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (1.49 + 3.60i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-9.85 - 9.85i)T + 71iT^{2} \) |
| 73 | \( 1 + (4.81 - 4.81i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + (1.15 + 2.77i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.82 + 6.82i)T - 89iT^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−9.551326627011104266288471438773, −8.726605577068961090607278222539, −8.047992926068981177358646869142, −7.22628876967316611464728130788, −6.52021112724221862488794706226, −5.14808637016770965941128539813, −4.32934005327139401557322151783, −3.76648631985359591964352354404, −2.11224837106373926827018839976, −0.75307325474334130571895553574,
1.29854327663433249874851259834, 2.86173077313544060450299250690, 3.57882534137092991945352569879, 5.01337481369559472271097956498, 5.46768621884273805283857166130, 6.65755701107531987777171531342, 7.74623483021578719217450058905, 8.141088130987922646824164356872, 8.935436182953788264294908865838, 10.03681074962058202508305724080