L(s) = 1 | + (−0.972 + 0.706i)2-s + (−0.309 − 0.951i)3-s + (−0.171 + 0.527i)4-s + (−1.69 − 1.23i)5-s + (0.972 + 0.706i)6-s + (−0.640 + 1.97i)7-s + (−0.949 − 2.92i)8-s + (−0.809 + 0.587i)9-s + 2.51·10-s + (3.15 − 1.01i)11-s + 0.555·12-s + (0.809 − 0.587i)13-s + (−0.769 − 2.36i)14-s + (−0.647 + 1.99i)15-s + (2.08 + 1.51i)16-s + (0.922 + 0.669i)17-s + ⋯ |
L(s) = 1 | + (−0.687 + 0.499i)2-s + (−0.178 − 0.549i)3-s + (−0.0857 + 0.263i)4-s + (−0.757 − 0.550i)5-s + (0.397 + 0.288i)6-s + (−0.242 + 0.744i)7-s + (−0.335 − 1.03i)8-s + (−0.269 + 0.195i)9-s + 0.796·10-s + (0.951 − 0.306i)11-s + 0.160·12-s + (0.224 − 0.163i)13-s + (−0.205 − 0.633i)14-s + (−0.167 + 0.514i)15-s + (0.522 + 0.379i)16-s + (0.223 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.743456 + 0.107613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.743456 + 0.107613i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.15 + 1.01i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.972 - 0.706i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (1.69 + 1.23i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.640 - 1.97i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (-0.922 - 0.669i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.10 - 3.40i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 8.83T + 23T^{2} \) |
| 29 | \( 1 + (-0.795 + 2.44i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.77 + 3.46i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.41 - 4.36i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.27 - 7.00i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 + (3.44 + 10.6i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.70 + 2.69i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.80 - 5.56i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 7.54i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 9.04T + 67T^{2} \) |
| 71 | \( 1 + (3.60 + 2.62i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.38 + 7.34i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.64 + 1.19i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.55 + 3.31i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + (-13.4 + 9.76i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.54851249104208386813281128718, −10.04701602069433510851569924876, −8.973228864902484751523143357534, −8.494782448310299588887678970018, −7.67530091443516560011279103122, −6.70766823384437561212564502465, −5.80873294945378745257177300421, −4.31091771515166420482159383570, −3.11141731629640903546378435361, −0.932340204146043758028936901997,
0.966239149922050835953435630128, 2.97424989398067211527771541242, 4.09842042175263562187929120293, 5.20295358156832115785379766994, 6.64123329171943257045388915209, 7.41701141617003646533806318187, 8.806711459970764621696733917845, 9.367307065600546332065509269740, 10.32988693712863454761711944588, 11.06876647387453219094237374994