L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (2.63 − 0.258i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.54 − 4.40i)11-s + (−0.258 − 0.965i)12-s + (2.02 + 2.02i)13-s + (−2.47 + 0.931i)14-s + (0.500 − 0.866i)16-s + (6.71 + 1.79i)17-s + (0.965 + 0.258i)18-s + (−1.79 + 3.11i)19-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (0.995 − 0.0978i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.767 − 1.32i)11-s + (−0.0747 − 0.278i)12-s + (0.561 + 0.561i)13-s + (−0.661 + 0.248i)14-s + (0.125 − 0.216i)16-s + (1.62 + 0.436i)17-s + (0.227 + 0.0610i)18-s + (−0.412 + 0.714i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.310712379\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310712379\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.258i)T \) |
good | 11 | \( 1 + (2.54 + 4.40i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.02 - 2.02i)T + 13iT^{2} \) |
| 17 | \( 1 + (-6.71 - 1.79i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.79 + 6.71i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 8.81iT - 29T^{2} \) |
| 31 | \( 1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.50 + 2.01i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.13iT - 41T^{2} \) |
| 43 | \( 1 + (7.84 - 7.84i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.482 - 1.79i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.879 - 0.235i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.762 + 2.84i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 73 | \( 1 + (-3.50 + 13.0i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (10.4 + 6.04i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.99 + 1.99i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.82 + 6.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 13.4i)T - 97iT^{2} \) |
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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
−9.768772496873504786491801164684, −8.561099806487382079896889572689, −8.062051529660516624981733128354, −7.73536049037109330428531965876, −6.19971124377585206391303297665, −5.94320091692189756479277221154, −4.55802981164666327866951099901, −3.22257855939889921943706085710, −1.98491626163244728814186247974, −0.810090566473514380567535742062,
1.37933693377526446738513164300, 2.60838513165142846153741377636, 3.72474901466590190035054137832, 4.99420266318259304799106571877, 5.51776436317783864865359004516, 7.08799305417129716055963236074, 7.73548143123283145022282454132, 8.417990555633842561587190693233, 9.328472735479407294111002759963, 10.11486328751399181263887996385