L(s) = 1 | + 3i·3-s + 9.15i·5-s − 27.4·7-s − 9·9-s − 20.5i·11-s + 32.0i·13-s − 27.4·15-s − 111.·17-s + 129. i·19-s − 82.2i·21-s − 9.16·23-s + 41.1·25-s − 27i·27-s + 41.0i·29-s + 187.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.818i·5-s − 1.48·7-s − 0.333·9-s − 0.562i·11-s + 0.683i·13-s − 0.472·15-s − 1.59·17-s + 1.56i·19-s − 0.854i·21-s − 0.0830·23-s + 0.329·25-s − 0.192i·27-s + 0.262i·29-s + 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.175967 + 0.753364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.175967 + 0.753364i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 - 9.15iT - 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 + 20.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 32.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 89.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 216. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 754. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 379. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 599. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−13.97688373945185344773868411510, −12.97885982329157132518997640069, −11.66360683247368478331866392769, −10.57152655876415591547269649038, −9.742735484149604615352361279147, −8.622449718738678300323171465422, −6.85526711233969870412937232531, −6.05078342732781860751208086385, −4.06619187001677496390527624704, −2.84130777782109793203754866122,
0.42649129458420919847436324333, 2.68575984375190588330567942750, 4.59911670497521363428999385902, 6.20005229814512931253778175358, 7.18925671613010752966546073129, 8.725230393358586954673344103640, 9.531020535778768993725954081761, 10.92951141871600687948614241603, 12.35895739174467119353599773459, 12.97460233467944982781914892342