L(s) = 1 | + 2-s + (−1.36 − 1.06i)3-s + 4-s + (−1.36 − 1.06i)6-s + (−0.294 − 2.62i)7-s + 8-s + (0.716 + 2.91i)9-s − 1.43i·11-s + (−1.36 − 1.06i)12-s − 4.73·13-s + (−0.294 − 2.62i)14-s + 16-s − 2.59i·17-s + (0.716 + 2.91i)18-s − 2.72i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.786 − 0.616i)3-s + 0.5·4-s + (−0.556 − 0.436i)6-s + (−0.111 − 0.993i)7-s + 0.353·8-s + (0.238 + 0.971i)9-s − 0.431i·11-s + (−0.393 − 0.308i)12-s − 1.31·13-s + (−0.0786 − 0.702i)14-s + 0.250·16-s − 0.630i·17-s + (0.168 + 0.686i)18-s − 0.625i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.158783211\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158783211\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.36 + 1.06i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.294 + 2.62i)T \) |
good | 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 + 2.59iT - 17T^{2} \) |
| 19 | \( 1 + 2.72iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.91iT - 31T^{2} \) |
| 37 | \( 1 + 2.39iT - 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 0.432iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.05iT - 61T^{2} \) |
| 67 | \( 1 - 9.82iT - 67T^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 7.58T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.02581969210253645311936138210, −8.564697581509155700054021400574, −7.37865005989218223747990178287, −7.11215992856359314263517185493, −6.17760562130162486758453261144, −5.13906967703460066300822775250, −4.55797496012994569346088295085, −3.26159944611593550510067415173, −1.99071106830388782994112523682, −0.42652629392203022099145822970,
1.95285649869060385836475227809, 3.19404456357144715520940100948, 4.31867214529942819704327014735, 5.08025018642502746869858838374, 5.83149823067812444701534092479, 6.55557653455979651687036000394, 7.59198412073129460183375901694, 8.685888080492619613479674771451, 9.821803434565240541002865436233, 10.09356769537834968933920265412