L(s) = 1 | + (−0.338 − 1.37i)2-s + 3-s + (−1.77 + 0.929i)4-s + 0.0609i·5-s + (−0.338 − 1.37i)6-s − 3.75·7-s + (1.87 + 2.11i)8-s + 9-s + (0.0837 − 0.0206i)10-s + 1.10·11-s + (−1.77 + 0.929i)12-s + 4.93i·13-s + (1.27 + 5.15i)14-s + 0.0609i·15-s + (2.27 − 3.29i)16-s + 6.72·17-s + ⋯ |
L(s) = 1 | + (−0.239 − 0.970i)2-s + 0.577·3-s + (−0.885 + 0.464i)4-s + 0.0272i·5-s + (−0.138 − 0.560i)6-s − 1.42·7-s + (0.662 + 0.748i)8-s + 0.333·9-s + (0.0264 − 0.00652i)10-s + 0.332·11-s + (−0.511 + 0.268i)12-s + 1.37i·13-s + (0.339 + 1.37i)14-s + 0.0157i·15-s + (0.568 − 0.822i)16-s + 1.62·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28875 - 0.0928688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28875 - 0.0928688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.338 + 1.37i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (6.62 + 4.80i)T \) |
good | 5 | \( 1 - 0.0609iT - 5T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 4.93iT - 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 1.13iT - 19T^{2} \) |
| 23 | \( 1 - 2.63iT - 23T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 - 0.470T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 1.53iT - 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 - 8.21iT - 47T^{2} \) |
| 53 | \( 1 - 2.81iT - 53T^{2} \) |
| 59 | \( 1 - 8.36iT - 59T^{2} \) |
| 61 | \( 1 + 4.04iT - 61T^{2} \) |
| 71 | \( 1 - 8.01iT - 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 + 0.0220iT - 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 3.14iT - 97T^{2} \) |
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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
−9.981952599349691809167750008747, −9.507562467587276206277398632801, −8.947096735753170706052316472097, −7.83606903156291285014707235835, −6.94394921669330893319930727183, −5.82095907939965038827970964900, −4.38770916975481680777208307957, −3.51173616399850174605962825542, −2.75585350739350622338908144707, −1.30987792082583937168570111222,
0.76369307194305425382900134419, 2.97154821736184309146323301152, 3.77384611061001282465502379588, 5.17108433576593701277941208596, 6.03124081931059122059441959144, 6.88697667275195398119878946256, 7.71138320889461690797469570620, 8.495355382242394568635511406353, 9.388323138660779397651661341575, 9.977314396002437616595633272167