L(s) = 1 | + (−0.226 − 0.392i)3-s + 2.54·7-s + (1.39 − 2.42i)9-s + 2.22·11-s + (3.51 − 6.08i)13-s + (1.27 + 2.21i)17-s + (−2.70 − 3.41i)19-s + (−0.575 − 0.997i)21-s + (−4.01 + 6.95i)23-s − 2.62·27-s + (0.941 − 1.63i)29-s + 5.98·31-s + (−0.503 − 0.872i)33-s − 2.86·37-s − 3.17·39-s + ⋯ |
L(s) = 1 | + (−0.130 − 0.226i)3-s + 0.961·7-s + (0.465 − 0.806i)9-s + 0.670·11-s + (0.973 − 1.68i)13-s + (0.310 + 0.537i)17-s + (−0.620 − 0.784i)19-s + (−0.125 − 0.217i)21-s + (−0.837 + 1.44i)23-s − 0.505·27-s + (0.174 − 0.302i)29-s + 1.07·31-s + (−0.0876 − 0.151i)33-s − 0.470·37-s − 0.509·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.041921916\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041921916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.70 + 3.41i)T \) |
good | 3 | \( 1 + (0.226 + 0.392i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + (-3.51 + 6.08i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.27 - 2.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (4.01 - 6.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.941 + 1.63i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 + (3.67 + 6.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.36 - 4.09i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.14 - 8.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.17 + 10.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.13 + 7.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.32 - 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.91 - 5.04i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.979908596535269049960722028139, −8.193456196204335618304094560373, −7.67548773134447867898408322484, −6.59237223993798687917538960186, −5.95463024400532498775611507088, −5.09403313293487293061482384511, −4.00299662258085121297155045586, −3.30797942139203424275878683151, −1.78472206673415593598038184121, −0.854166870613623228559787421149,
1.42456140929434165814599837784, 2.16337488967381672117901750139, 3.78991619708252612792241069721, 4.45617688748083034123012595384, 5.09409729381329876624722848424, 6.39051525825552787876396257025, 6.76697586857695187367473735624, 8.133278227921016298696138682538, 8.324546006386704404916160599536, 9.366297589931398000332583341097