L(s) = 1 | − 3.23·2-s − 1.73·3-s + 6.47·4-s + 4.61i·5-s + 5.60·6-s − 2.64i·7-s − 8.01·8-s + 2.99·9-s − 14.9i·10-s − 2.93i·11-s − 11.2·12-s − 15.4·13-s + 8.56i·14-s − 7.99i·15-s + 0.0291·16-s + 24.9i·17-s + ⋯ |
L(s) = 1 | − 1.61·2-s − 0.577·3-s + 1.61·4-s + 0.923i·5-s + 0.934·6-s − 0.377i·7-s − 1.00·8-s + 0.333·9-s − 1.49i·10-s − 0.266i·11-s − 0.934·12-s − 1.18·13-s + 0.611i·14-s − 0.533i·15-s + 0.00182·16-s + 1.46i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1195764434\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1195764434\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (-19.6 - 11.9i)T \) |
good | 2 | \( 1 + 3.23T + 4T^{2} \) |
| 5 | \( 1 - 4.61iT - 25T^{2} \) |
| 11 | \( 1 + 2.93iT - 121T^{2} \) |
| 13 | \( 1 + 15.4T + 169T^{2} \) |
| 17 | \( 1 - 24.9iT - 289T^{2} \) |
| 19 | \( 1 + 15.1iT - 361T^{2} \) |
| 29 | \( 1 + 2.82T + 841T^{2} \) |
| 31 | \( 1 + 9.30T + 961T^{2} \) |
| 37 | \( 1 - 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.66iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 1.59iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 94.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 115. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 65.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 5.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 54.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 172. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 1.81iT - 9.40e3T^{2} \) |
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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
−10.48328395349112259666807881916, −9.724972329415934415723384353276, −8.766446688827032899678081396628, −7.69362532452312819379641023529, −7.02376973307292308838724361013, −6.30722990830948509234176516920, −4.81625246816995058628286623678, −3.12912312633179712891783003668, −1.69656815063480855121302010087, −0.10348657903527215713136580107,
1.11796014443274395457493916656, 2.50055634997591473477367190963, 4.63050595317824998205716063481, 5.46960280539578381360199516209, 6.91959069707721360376280062963, 7.51523603549545145414990813934, 8.639426437632225984565449943853, 9.262970233811208070557111588637, 9.979525783792771613210724144988, 10.80889701992616042378278321736