L(s) = 1 | + (0.366 + 1.36i)2-s + (−2.45 + 1.72i)3-s + (−1.73 + i)4-s + (2.65 − 4.23i)5-s + (−3.25 − 2.72i)6-s + (3.47 − 6.07i)7-s + (−2 − 1.99i)8-s + (3.04 − 8.46i)9-s + (6.75 + 2.07i)10-s + (5.23 − 3.02i)11-s + (2.52 − 5.44i)12-s + (7.33 + 7.33i)13-s + (9.57 + 2.51i)14-s + (0.787 + 14.9i)15-s + (1.99 − 3.46i)16-s + (6.03 − 22.5i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.818 + 0.575i)3-s + (−0.433 + 0.250i)4-s + (0.531 − 0.847i)5-s + (−0.542 − 0.453i)6-s + (0.495 − 0.868i)7-s + (−0.250 − 0.249i)8-s + (0.338 − 0.940i)9-s + (0.675 + 0.207i)10-s + (0.475 − 0.274i)11-s + (0.210 − 0.453i)12-s + (0.564 + 0.564i)13-s + (0.683 + 0.179i)14-s + (0.0524 + 0.998i)15-s + (0.124 − 0.216i)16-s + (0.354 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41847 + 0.158399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41847 + 0.158399i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 3 | \( 1 + (2.45 - 1.72i)T \) |
| 5 | \( 1 + (-2.65 + 4.23i)T \) |
| 7 | \( 1 + (-3.47 + 6.07i)T \) |
good | 11 | \( 1 + (-5.23 + 3.02i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.33 - 7.33i)T + 169iT^{2} \) |
| 17 | \( 1 + (-6.03 + 22.5i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (17.0 - 29.6i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-26.5 + 7.11i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 6.86T + 841T^{2} \) |
| 31 | \( 1 + (-38.9 + 22.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-34.1 + 9.14i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 18.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-17.2 + 17.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (23.3 - 6.24i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-20.7 + 77.2i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (97.9 - 56.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-16.8 - 9.74i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.92 - 10.9i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 80.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (0.519 - 1.93i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-73.1 - 42.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-4.56 + 4.56i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (21.5 + 12.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (38.4 - 38.4i)T - 9.40e3iT^{2} \) |
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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
−12.16033342462045084122988046228, −11.26416604830300289623920631765, −10.11471952569450483529252388134, −9.253307638843028863015272035853, −8.187635405429352608862910496494, −6.79755544622769649884845678941, −5.85434736371424219473060325500, −4.78700337208283945195784167248, −3.97451781755268286931556522093, −0.986465525332442916081941819871,
1.52573815029914892009946866006, 2.83268201815007338197663145794, 4.72655599379931831251934725065, 5.89273114090757063592517955352, 6.66036867723748683190644349204, 8.134839069335552729375495255788, 9.332303168759996480227909351932, 10.69997912382427548373610388314, 11.01193617553932635143160744206, 12.06002904068032012567205994402