L(s) = 1 | + (−1.69 + 0.335i)3-s − 3.64·5-s + (1.05 + 2.42i)7-s + (2.77 − 1.14i)9-s − 1.39i·11-s + (−2.97 − 1.71i)13-s + (6.19 − 1.22i)15-s + (−2.41 + 4.18i)17-s + (7.23 − 4.17i)19-s + (−2.60 − 3.76i)21-s − 8.92i·23-s + 8.29·25-s + (−4.33 + 2.86i)27-s + (5.02 − 2.89i)29-s + (5.29 − 3.05i)31-s + ⋯ |
L(s) = 1 | + (−0.981 + 0.193i)3-s − 1.63·5-s + (0.398 + 0.917i)7-s + (0.924 − 0.380i)9-s − 0.421i·11-s + (−0.825 − 0.476i)13-s + (1.59 − 0.315i)15-s + (−0.586 + 1.01i)17-s + (1.66 − 0.958i)19-s + (−0.568 − 0.822i)21-s − 1.86i·23-s + 1.65·25-s + (−0.833 + 0.552i)27-s + (0.932 − 0.538i)29-s + (0.951 − 0.549i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.520885 - 0.282171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.520885 - 0.282171i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 - 0.335i)T \) |
| 7 | \( 1 + (-1.05 - 2.42i)T \) |
good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 11 | \( 1 + 1.39iT - 11T^{2} \) |
| 13 | \( 1 + (2.97 + 1.71i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.41 - 4.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.23 + 4.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.92iT - 23T^{2} \) |
| 29 | \( 1 + (-5.02 + 2.89i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.29 + 3.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.89 + 5.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.802 + 1.38i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.51 + 2.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.40 + 2.54i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.21 - 5.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.78 + 4.49i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.53 - 4.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.20iT - 71T^{2} \) |
| 73 | \( 1 + (0.745 + 0.430i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.31 - 2.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.82 + 6.62i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.44 + 7.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.63 + 5.56i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.05575505994455652839439902681, −10.12390687420909986376134285586, −8.853248020415212978853474104098, −8.056740320992852354590224380680, −7.15384956034626846591648483575, −6.08928992310702379049720962147, −4.94228100864966871636663609571, −4.27913971194090435305058795211, −2.84323561560875197325762836395, −0.49330012204480889246654299454,
1.15040893457989051429532415504, 3.42929254983215444815298991371, 4.53581926636644348156941735534, 5.11543735125966145559650445646, 6.81168040447559034503303138578, 7.42845101892018253150850542582, 7.88608961152710000265050574017, 9.478697358189137028999142286861, 10.35819920650418017032162215686, 11.35019453652944132332052369399