| L(s) = 1 | + (−2.04 + 0.324i)3-s + (1.25 + 1.85i)5-s + (1.74 + 1.74i)7-s + (1.22 − 0.399i)9-s + (−3.97 − 1.29i)11-s + (1.72 − 3.39i)13-s + (−3.16 − 3.38i)15-s + (−0.973 + 6.14i)17-s + (−5.07 + 3.68i)19-s + (−4.13 − 3.00i)21-s + (1.78 + 3.49i)23-s + (−1.86 + 4.63i)25-s + (3.15 − 1.60i)27-s + (−5.30 + 7.30i)29-s + (1.52 + 2.10i)31-s + ⋯ |
| L(s) = 1 | + (−1.18 + 0.187i)3-s + (0.559 + 0.828i)5-s + (0.658 + 0.658i)7-s + (0.409 − 0.133i)9-s + (−1.19 − 0.389i)11-s + (0.479 − 0.941i)13-s + (−0.816 − 0.874i)15-s + (−0.236 + 1.49i)17-s + (−1.16 + 0.845i)19-s + (−0.901 − 0.654i)21-s + (0.371 + 0.729i)23-s + (−0.373 + 0.927i)25-s + (0.606 − 0.309i)27-s + (−0.985 + 1.35i)29-s + (0.274 + 0.377i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.346736 + 0.658591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.346736 + 0.658591i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.25 - 1.85i)T \) |
| good | 3 | \( 1 + (2.04 - 0.324i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-1.74 - 1.74i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.97 + 1.29i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.72 + 3.39i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.973 - 6.14i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (5.07 - 3.68i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.78 - 3.49i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (5.30 - 7.30i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.52 - 2.10i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.92 + 1.99i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (2.43 + 7.49i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.559 + 0.559i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.158 - 0.998i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.0860 - 0.543i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (0.0466 + 0.143i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.875 - 2.69i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-12.5 - 1.98i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-2.76 + 3.81i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.57 + 2.33i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-1.99 - 1.45i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.40 - 8.84i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-15.7 - 5.12i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (10.0 - 1.59i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.25349109139030440783867397733, −10.59520163053933364332164747091, −10.42921774826179702684142139275, −8.751837998888443857327274748342, −7.930467811818399376192049958506, −6.56771255390100688355414052727, −5.60543166380481575600441835508, −5.34652595827967472489519921654, −3.52212544820493466148742244687, −2.00041587622286683119443151966,
0.54522896438562518184705393273, 2.20764672503044478770884506915, 4.58350442620306076852605591166, 4.93456953849621641575026290298, 6.12359813122777822420114272453, 7.01666030536767440575803040942, 8.168425556034207226909858062171, 9.201346989957127374590767228983, 10.22363154339684272426346137478, 11.15882079466356719836980793414
