| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 8.12i·5-s + (−4 − 5.74i)7-s + 2.82·8-s − 11.4i·10-s + 12.7·11-s + 22.9i·13-s + (−5.65 − 8.12i)14-s + 4.00·16-s − 8.12i·17-s + 11.4i·19-s − 16.2i·20-s + 18·22-s + 21.2·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 1.62i·5-s + (−0.571 − 0.820i)7-s + 0.353·8-s − 1.14i·10-s + 1.15·11-s + 1.76i·13-s + (−0.404 − 0.580i)14-s + 0.250·16-s − 0.477i·17-s + 0.604i·19-s − 0.812i·20-s + 0.818·22-s + 0.922·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.79659 - 0.938239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79659 - 0.938239i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4 + 5.74i)T \) |
| good | 5 | \( 1 + 8.12iT - 25T^{2} \) |
| 11 | \( 1 - 12.7T + 121T^{2} \) |
| 13 | \( 1 - 22.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.12iT - 289T^{2} \) |
| 19 | \( 1 - 11.4iT - 361T^{2} \) |
| 23 | \( 1 - 21.2T + 529T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 - 56.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 22.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 52T + 4.48e3T^{2} \) |
| 71 | \( 1 + 89.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 45.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 104T + 6.24e3T^{2} \) |
| 83 | \( 1 + 162. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 91.9iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.04555630612934453115113230607, −12.11447462575265414432558488673, −11.31311819164470135216206232857, −9.590304122254958674214760967805, −8.954999517266684480301556938703, −7.33827580467804692197832638757, −6.19767625153465680355170461673, −4.68105477127415839002847091602, −3.90956218647474449240570523247, −1.37417304224821295229264961243,
2.66914301156086103093288430329, 3.59467404789706305638789990584, 5.64196061328522157600082960094, 6.50257469335578455465174726993, 7.54122839384197865419772652932, 9.208590479469739543456344317057, 10.49286184465069760413311893221, 11.22325764982783746230019264376, 12.36622332664861936918865051745, 13.27456662299126233257501249014
