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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4842666906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4842666906\) |
\(L(\frac{3}{2})\) |
|
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
−9.657850001975868934820901535449, −8.887232066219535193491669963536, −7.85425985668114292972382127050, −7.30948559596447910548214697672, −6.25712241425748167431156801513, −5.63265859002482984259960665048, −4.67676941304443642555187551119, −2.86829857089241342578859911533, −1.96244441624210368675743243250, −0.34763995613045536531030086021,
1.12018324048485312692200148944, 2.83882082295117982151459478679, 4.14809229960029952903585559467, 4.89886090946972697518736162693, 5.99294083407951804045759581300, 7.07552388893988077869111669105, 7.37415549699592195748590625294, 8.734649679809390205269404993625, 9.553657358273284381074343483640, 10.14947705242145874575983958703