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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.586148038\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.586148038\) |
\(L(\frac{3}{2})\) |
|
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
−9.941199326045672760584235255574, −9.092234981818155872606067066240, −8.546993428648783030868644725650, −8.120797454135003462482337845221, −6.84621612603644066236504737532, −5.90531305409683522622835388904, −4.91697467779775207020266214438, −3.61182027345787771849263636128, −2.78370733915314456408522491765, −1.56950214996938336022135897936,
0.889310530798013407224923511455, 1.93320622404050892587663754646, 3.28891950587675926195582837574, 4.10334736527506044688966067542, 5.64766732751926810226247441558, 6.89523843281334890586365273071, 7.14073208549732333503132731907, 8.236409683673270187857838638490, 8.661463729414643380276327085316, 9.708885631201256572308053402472