L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.189 + 2.63i)7-s − 0.999·8-s + (−1.32 − 0.766i)11-s − 1.48·13-s + (−2.19 + 1.48i)14-s + (−0.5 − 0.866i)16-s + (2.10 + 1.21i)17-s + (−4.21 + 2.43i)19-s − 1.53i·22-s + (−0.133 − 0.232i)23-s + (−0.741 − 1.28i)26-s + (−2.38 − 1.15i)28-s − 0.898i·29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0716 + 0.997i)7-s − 0.353·8-s + (−0.400 − 0.230i)11-s − 0.411·13-s + (−0.585 + 0.396i)14-s + (−0.125 − 0.216i)16-s + (0.510 + 0.294i)17-s + (−0.966 + 0.557i)19-s − 0.326i·22-s + (−0.0279 − 0.0483i)23-s + (−0.145 − 0.251i)26-s + (−0.449 − 0.218i)28-s − 0.166i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4757454367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4757454367\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.189 - 2.63i)T \) |
good | 11 | \( 1 + (1.32 + 0.766i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + (-2.10 - 1.21i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.21 - 2.43i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.133 + 0.232i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (0.717 + 0.414i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.74 - 2.74i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.76T + 41T^{2} \) |
| 43 | \( 1 + 1.86iT - 43T^{2} \) |
| 47 | \( 1 + (-6.46 + 3.72i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.73 + 3.00i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.12 - 5.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.73 + 3.31i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.8 + 8.01i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (-0.171 + 0.297i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.22 + 9.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.45iT - 83T^{2} \) |
| 89 | \( 1 + (7.98 + 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.771792129853909060920662610966, −8.488983911449326369152498948632, −7.66582443705115884985033693934, −6.84549564635507455216753937968, −6.01027431100877783281277140211, −5.47699312942237990194355447968, −4.71662522717045056673386692317, −3.72483544017149815098422462632, −2.81143098735057610283676889671, −1.82299298221926028083235423501,
0.12405743037466701370456116708, 1.40986879871683962213407304302, 2.49174462520679842273035628936, 3.41024905765645980818057340139, 4.28808298300726631902198736517, 4.90701777014296741962000139728, 5.76272821286160704832224618727, 6.80939389258504457924637868804, 7.34770265706349466822307633002, 8.235466744752119998768525828149