L(s) = 1 | + 7-s + (3.5 + 6.06i)13-s + (−4 + 1.73i)19-s + (2.5 + 4.33i)25-s − 11·31-s − 37-s + (2.5 − 4.33i)43-s − 6·49-s + (6.5 + 11.2i)61-s + (2.5 + 4.33i)67-s + (3.5 − 6.06i)73-s + (−6.5 + 11.2i)79-s + (3.5 + 6.06i)91-s + (−7 + 12.1i)97-s + 13·103-s + ⋯ |
L(s) = 1 | + 0.377·7-s + (0.970 + 1.68i)13-s + (−0.917 + 0.397i)19-s + (0.5 + 0.866i)25-s − 1.97·31-s − 0.164·37-s + (0.381 − 0.660i)43-s − 0.857·49-s + (0.832 + 1.44i)61-s + (0.305 + 0.529i)67-s + (0.409 − 0.709i)73-s + (−0.731 + 1.26i)79-s + (0.366 + 0.635i)91-s + (−0.710 + 1.23i)97-s + 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.486755136\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.486755136\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.909651498756595813186719366062, −8.481380592290244429273515498935, −7.40441181731652690071841014593, −6.79812311692226590575953247065, −5.99249343042853713846521599854, −5.14904996723655646782918689430, −4.15856331671155656265568194913, −3.61025810929865884953444879209, −2.18232044308614782165028871657, −1.40860371051596851626892960750,
0.48344629813890547028717724100, 1.78167645699854059480313143426, 2.93321309807980855637419691174, 3.75754052939657342197403457249, 4.75148266802816772793363610123, 5.55845607058151059496893826361, 6.25126405539364527328459067714, 7.14050091451346010672859184291, 8.067176148519715323875600612147, 8.443902891164979404425085391891