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
−8.235466744752119998768525828149, −7.34770265706349466822307633002, −6.80939389258504457924637868804, −5.76272821286160704832224618727, −4.90701777014296741962000139728, −4.28808298300726631902198736517, −3.41024905765645980818057340139, −2.49174462520679842273035628936, −1.40986879871683962213407304302, −0.12405743037466701370456116708,
1.82299298221926028083235423501, 2.81143098735057610283676889671, 3.72483544017149815098422462632, 4.71662522717045056673386692317, 5.47699312942237990194355447968, 6.01027431100877783281277140211, 6.84549564635507455216753937968, 7.66582443705115884985033693934, 8.488983911449326369152498948632, 8.771792129853909060920662610966