L(s) = 1 | − 2.48·2-s + (0.809 + 1.53i)3-s + 4.17·4-s − 2.08i·5-s + (−2.01 − 3.80i)6-s + 0.856i·7-s − 5.41·8-s + (−1.69 + 2.47i)9-s + 5.18i·10-s + (3.26 − 0.569i)11-s + (3.38 + 6.39i)12-s + i·13-s − 2.12i·14-s + (3.19 − 1.68i)15-s + 5.10·16-s + 3.81·17-s + ⋯ |
L(s) = 1 | − 1.75·2-s + (0.467 + 0.884i)3-s + 2.08·4-s − 0.932i·5-s + (−0.821 − 1.55i)6-s + 0.323i·7-s − 1.91·8-s + (−0.563 + 0.826i)9-s + 1.63i·10-s + (0.985 − 0.171i)11-s + (0.975 + 1.84i)12-s + 0.277i·13-s − 0.569i·14-s + (0.824 − 0.435i)15-s + 1.27·16-s + 0.926·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.722782 + 0.247036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.722782 + 0.247036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 1.53i)T \) |
| 11 | \( 1 + (-3.26 + 0.569i)T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 2.08iT - 5T^{2} \) |
| 7 | \( 1 - 0.856iT - 7T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 + 4.36iT - 19T^{2} \) |
| 23 | \( 1 - 3.17iT - 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 - 9.59T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 8.13iT - 43T^{2} \) |
| 47 | \( 1 - 4.07iT - 47T^{2} \) |
| 53 | \( 1 + 1.21iT - 53T^{2} \) |
| 59 | \( 1 + 4.04iT - 59T^{2} \) |
| 61 | \( 1 - 2.58iT - 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.49iT - 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 15.4iT - 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−11.10955481191435073278933574511, −9.776888553835849274691222914027, −9.503738205508548817190506014162, −8.680347756814904802508057677557, −8.153050876863168592291964007531, −6.98398387854978896610041355615, −5.64416852548016688363705776158, −4.33485715282855008532658679589, −2.78194312414188715809274310898, −1.20890449281072156192098102963,
1.04067795190397105695374690405, 2.32699543278853826206255223751, 3.53970272694521408150118703863, 6.10229389337015271322847822138, 6.82783947226626948191966202957, 7.54378930164860386408810583713, 8.267240158314001860147813396284, 9.188200960819163251455491181087, 10.06409828678851622030652646167, 10.76369430119659483655097966540