L(s) = 1 | + 2-s + 4-s + (1 − 1.73i)5-s + 8-s + (1 − 1.73i)10-s + (0.5 + 0.866i)11-s + (3 + 5.19i)13-s + 16-s + (−2.5 + 4.33i)17-s + (3.5 + 6.06i)19-s + (1 − 1.73i)20-s + (0.5 + 0.866i)22-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + (3 + 5.19i)26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.447 − 0.774i)5-s + 0.353·8-s + (0.316 − 0.547i)10-s + (0.150 + 0.261i)11-s + (0.832 + 1.44i)13-s + 0.250·16-s + (−0.606 + 1.05i)17-s + (0.802 + 1.39i)19-s + (0.223 − 0.387i)20-s + (0.106 + 0.184i)22-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + (0.588 + 1.01i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.259559663\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.259559663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + (-8 + 13.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.917983940601465354939741345583, −8.266326861712310932284682929211, −7.18847173992713337705876950338, −6.48426227974975914469126083330, −5.74800514234666326904050439052, −5.02337073875204599186372971155, −4.10036243895398022835611411951, −3.54758530838991231870481696749, −1.99728705025649764464269649188, −1.43025321675836320593810492964,
0.885080629689844153505684757162, 2.45039758087918415575607234136, 3.02792937841832216522233550271, 3.87245379280636323075064588547, 5.09183853714789162143657185009, 5.58957686407212049532789372439, 6.46198629453036784790016244634, 7.13654956025678900262641677957, 7.80115269011844376530276373581, 8.894016847431953400980822889631