| L(s) = 1 | + (−0.544 + 0.838i)2-s + (−1.15 + 3.01i)3-s + (−0.406 − 0.913i)4-s + (−1.77 − 1.35i)5-s + (−1.89 − 2.61i)6-s + (−0.00275 + 2.64i)7-s + (0.987 + 0.156i)8-s + (−5.51 − 4.96i)9-s + (2.10 − 0.755i)10-s + (0.696 + 0.773i)11-s + (3.22 − 0.168i)12-s + (−3.97 − 2.02i)13-s + (−2.21 − 1.44i)14-s + (6.13 − 3.79i)15-s + (−0.669 + 0.743i)16-s + (1.70 − 2.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.385 + 0.593i)2-s + (−0.667 + 1.73i)3-s + (−0.203 − 0.456i)4-s + (−0.795 − 0.605i)5-s + (−0.774 − 1.06i)6-s + (−0.00104 + 0.999i)7-s + (0.349 + 0.0553i)8-s + (−1.83 − 1.65i)9-s + (0.665 − 0.238i)10-s + (0.209 + 0.233i)11-s + (0.930 − 0.0487i)12-s + (−1.10 − 0.561i)13-s + (−0.592 − 0.385i)14-s + (1.58 − 0.980i)15-s + (−0.167 + 0.185i)16-s + (0.413 − 0.511i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0700047 - 0.0586595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0700047 - 0.0586595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.544 - 0.838i)T \) |
| 5 | \( 1 + (1.77 + 1.35i)T \) |
| 7 | \( 1 + (0.00275 - 2.64i)T \) |
| good | 3 | \( 1 + (1.15 - 3.01i)T + (-2.22 - 2.00i)T^{2} \) |
| 11 | \( 1 + (-0.696 - 0.773i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.97 + 2.02i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 2.10i)T + (-3.53 - 16.6i)T^{2} \) |
| 19 | \( 1 + (2.97 + 1.32i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-4.00 - 2.60i)T + (9.35 + 21.0i)T^{2} \) |
| 29 | \( 1 + (-2.96 + 4.08i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.23 - 0.549i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.133 - 2.53i)T + (-36.7 + 3.86i)T^{2} \) |
| 41 | \( 1 + (-1.71 + 0.557i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.92 + 4.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.14 - 7.40i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (6.25 + 2.40i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (13.2 + 2.80i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (0.578 + 2.72i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (4.15 + 3.36i)T + (13.9 + 65.5i)T^{2} \) |
| 71 | \( 1 + (-6.39 - 4.64i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (16.7 + 0.879i)T + (72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (11.8 + 1.24i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.547 + 3.45i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-11.4 + 2.42i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (-0.878 - 5.54i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.88054742081804796504503623553, −11.23846623402128261196755322419, −10.14725043458943666313027721746, −9.364181468533614644073707779456, −8.805326047871810829431282714418, −7.66524004655836102172887498208, −6.17230211840065364006316799438, −5.04786163904439903820782469269, −4.74494528545006123856476865133, −3.24147743230796426528841640433,
0.07778146214850433535418617473, 1.60878243805888021260494004311, 3.08546216770310709635132047572, 4.63326642166390047060678232250, 6.34278629567640490124703420824, 7.13263605156622853191443932119, 7.65952308503255335308774332254, 8.619464159362834477577308007939, 10.26433937479522319779746873933, 11.01945178170454200580767516878
