| L(s) = 1 | + (2.82 + i)3-s + 5.65·5-s + 6·7-s + (7.00 + 5.65i)9-s + 5.65·11-s + 10i·13-s + (16.0 + 5.65i)15-s − 22.6i·17-s + 2i·19-s + (16.9 + 6i)21-s + 11.3i·23-s + 7.00·25-s + (14.1 + 23.0i)27-s − 16.9·29-s − 22·31-s + ⋯ |
| L(s) = 1 | + (0.942 + 0.333i)3-s + 1.13·5-s + 0.857·7-s + (0.777 + 0.628i)9-s + 0.514·11-s + 0.769i·13-s + (1.06 + 0.377i)15-s − 1.33i·17-s + 0.105i·19-s + (0.808 + 0.285i)21-s + 0.491i·23-s + 0.280·25-s + (0.523 + 0.851i)27-s − 0.585·29-s − 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.639081119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.639081119\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 - i)T \) |
| good | 5 | \( 1 - 5.65T + 25T^{2} \) |
| 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 - 5.65T + 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 - 2iT - 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 16.9T + 841T^{2} \) |
| 31 | \( 1 + 22T + 961T^{2} \) |
| 37 | \( 1 - 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 86iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−9.870311590646599238999239815510, −9.347558579269011861980572274050, −8.728035157732152876564029016949, −7.62594343129899795198019216854, −6.86159388699743650356630626279, −5.57512184067441829725147637285, −4.74996645028537781602980107507, −3.66347307191241547862920928506, −2.32179501457604473347131766545, −1.56810362087848676518952423756,
1.37351905895664803529774411579, 2.12188415614471564829359487149, 3.38101891999286757384316797068, 4.54670303268171059139487069080, 5.75376409033570148714038904854, 6.52636106364771798566693373061, 7.68413579349529989931051792368, 8.354355296185594209711140475857, 9.153636851879064898953267101763, 9.954470371227011567345531849633
