L(s) = 1 | + (0.171 + 0.297i)5-s + (0.271 + 2.63i)7-s + (2.45 − 4.26i)11-s + (0.974 − 1.68i)13-s + (−2.07 − 3.59i)17-s + (0.202 − 0.350i)19-s + (1.37 + 2.38i)23-s + (2.44 − 4.22i)25-s + (−1.63 − 2.83i)29-s − 3.28·31-s + (−0.736 + 0.533i)35-s + (−3.38 + 5.87i)37-s + (2.81 − 4.87i)41-s + (−4.96 − 8.59i)43-s + 13.2·47-s + ⋯ |
L(s) = 1 | + (0.0768 + 0.133i)5-s + (0.102 + 0.994i)7-s + (0.741 − 1.28i)11-s + (0.270 − 0.467i)13-s + (−0.502 − 0.871i)17-s + (0.0464 − 0.0803i)19-s + (0.287 + 0.498i)23-s + (0.488 − 0.845i)25-s + (−0.304 − 0.527i)29-s − 0.590·31-s + (−0.124 + 0.0901i)35-s + (−0.557 + 0.965i)37-s + (0.439 − 0.760i)41-s + (−0.756 − 1.31i)43-s + 1.92·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.786545092\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786545092\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.271 - 2.63i)T \) |
good | 5 | \( 1 + (-0.171 - 0.297i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.45 + 4.26i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.974 + 1.68i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.07 + 3.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.202 + 0.350i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.63 + 2.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + (3.38 - 5.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 4.87i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.96 + 8.59i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + (1.38 + 2.39i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + (-4.58 - 7.93i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + (4.62 + 8.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.54 - 6.13i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.31 + 2.27i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.688111576768916799139645091701, −8.560278740251796132056002038290, −7.33135047685179750528838335727, −6.53890948645803853405731401605, −5.73317539622918027831638036639, −5.18265073996543357696118258577, −3.93888057632287602864966744279, −3.07435463897296238537958186059, −2.16336160840658326891050112362, −0.69356706965403858913634434669,
1.19615059776040497359223367291, 2.08183657085976748906796313744, 3.58574363498050015768396848916, 4.23427495120076260789963789107, 4.95509281892594688748952962277, 6.10979748175727195119042072546, 6.97621093116188262255669733253, 7.32226159652310562367095629326, 8.400647599626878801474803377235, 9.173974153496262272915379951141