L(s) = 1 | + (−1.22 + 1.22i)3-s + (−4.22 − 2.67i)5-s + (−7.44 − 7.44i)7-s − 2.99i·9-s − 16.2·11-s + (−12.2 + 12.2i)13-s + (8.44 − 1.89i)15-s + (−7.55 − 7.55i)17-s − 14.4i·19-s + 18.2·21-s + (2.65 − 2.65i)23-s + (10.6 + 22.5i)25-s + (3.67 + 3.67i)27-s + 34.2i·29-s + 20.4·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.844 − 0.534i)5-s + (−1.06 − 1.06i)7-s − 0.333i·9-s − 1.47·11-s + (−0.942 + 0.942i)13-s + (0.563 − 0.126i)15-s + (−0.444 − 0.444i)17-s − 0.762i·19-s + 0.868·21-s + (0.115 − 0.115i)23-s + (0.427 + 0.903i)25-s + (0.136 + 0.136i)27-s + 1.18i·29-s + 0.661·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3835059754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3835059754\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (4.22 + 2.67i)T \) |
good | 7 | \( 1 + (7.44 + 7.44i)T + 49iT^{2} \) |
| 11 | \( 1 + 16.2T + 121T^{2} \) |
| 13 | \( 1 + (12.2 - 12.2i)T - 169iT^{2} \) |
| 17 | \( 1 + (7.55 + 7.55i)T + 289iT^{2} \) |
| 19 | \( 1 + 14.4iT - 361T^{2} \) |
| 23 | \( 1 + (-2.65 + 2.65i)T - 529iT^{2} \) |
| 29 | \( 1 - 34.2iT - 841T^{2} \) |
| 31 | \( 1 - 20.4T + 961T^{2} \) |
| 37 | \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 25.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-25.1 + 25.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (22.0 + 22.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-35.3 + 35.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 88.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (24.6 + 24.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 77.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (44.1 - 44.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 48.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (101. - 101. i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 156. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-55.4 - 55.4i)T + 9.40e3iT^{2} \) |
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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
−9.954224086896994405688583987932, −9.261434242365753895549224277830, −8.266417629553996786340777285134, −7.11932156028129243263715104074, −6.89954371147042518681780433122, −5.31427225111918673433438744652, −4.64965687854508839130360508695, −3.78842580636177786052440646301, −2.66709626725209588552881360816, −0.56595540492996294178206128458,
0.24286276559198352858481679676, 2.50492560865685077733631763834, 2.99796260948533915400665607668, 4.45299451437637856448790755549, 5.64248186464790399114159692443, 6.15502557500847672351876620000, 7.34087935528025081593583689683, 7.86697954484995763904318418452, 8.743334780518776432311579433128, 10.08104069756196771109648604288