| L(s) = 1 | + (4.57 − 3.32i)2-s + (15.4 − 21.2i)3-s + (9.88 − 30.4i)4-s + 6.26·5-s − 148. i·6-s + (109. − 336. i)7-s + (−55.9 − 172. i)8-s + (12.0 + 36.9i)9-s + (28.6 − 20.8i)10-s + (−988. − 321. i)11-s + (−494. − 680. i)12-s + (1.59e3 − 2.20e3i)13-s + (−619. − 1.90e3i)14-s + (96.7 − 133. i)15-s + (−828. − 601. i)16-s + (576. − 187. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.571 − 0.787i)3-s + (0.154 − 0.475i)4-s + 0.0501·5-s − 0.687i·6-s + (0.319 − 0.982i)7-s + (−0.109 − 0.336i)8-s + (0.0164 + 0.0507i)9-s + (0.0286 − 0.0208i)10-s + (−0.742 − 0.241i)11-s + (−0.285 − 0.393i)12-s + (0.727 − 1.00i)13-s + (−0.225 − 0.694i)14-s + (0.0286 − 0.0394i)15-s + (−0.202 − 0.146i)16-s + (0.117 − 0.0381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.32240 - 2.58363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32240 - 2.58363i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.57 + 3.32i)T \) |
| 31 | \( 1 + (-9.74e3 + 2.81e4i)T \) |
| good | 3 | \( 1 + (-15.4 + 21.2i)T + (-225. - 693. i)T^{2} \) |
| 5 | \( 1 - 6.26T + 1.56e4T^{2} \) |
| 7 | \( 1 + (-109. + 336. i)T + (-9.51e4 - 6.91e4i)T^{2} \) |
| 11 | \( 1 + (988. + 321. i)T + (1.43e6 + 1.04e6i)T^{2} \) |
| 13 | \( 1 + (-1.59e3 + 2.20e3i)T + (-1.49e6 - 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-576. + 187. i)T + (1.95e7 - 1.41e7i)T^{2} \) |
| 19 | \( 1 + (9.17e3 - 6.66e3i)T + (1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 + (-1.29e4 + 4.21e3i)T + (1.19e8 - 8.70e7i)T^{2} \) |
| 29 | \( 1 + (3.01e3 + 4.15e3i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 37 | \( 1 + 2.07e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-2.16e4 + 1.57e4i)T + (1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + (-7.81e4 - 1.07e5i)T + (-1.95e9 + 6.01e9i)T^{2} \) |
| 47 | \( 1 + (-6.99e4 - 5.07e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-1.65e5 + 5.37e4i)T + (1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-2.13e5 - 1.55e5i)T + (1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 - 2.40e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.35e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-2.45e4 - 7.55e4i)T + (-1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-8.78e4 - 2.85e4i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-2.06e5 + 6.72e4i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-2.35e4 - 3.24e4i)T + (-1.01e11 + 3.10e11i)T^{2} \) |
| 89 | \( 1 + (1.25e6 + 4.07e5i)T + (4.02e11 + 2.92e11i)T^{2} \) |
| 97 | \( 1 + (2.13e4 - 6.57e4i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
| 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
−13.26642523236537517202019221970, −12.73672410606042798691496751844, −11.03552855827538062866633817528, −10.28551890376827974174412679062, −8.335966887330448707972651025800, −7.43354764886522671440434687063, −5.83607886145991125475300289969, −4.10580545306369925540593499463, −2.50620056859599533661486080593, −0.973100217448897568447663358743,
2.42686131004298534303053109252, 3.98352730515254990189704603736, 5.23918563349253533662254414510, 6.73519304776810970491265730920, 8.473851911289850657898132712573, 9.212736035659905212249480661874, 10.78401401270839940960944790871, 12.04516599614264391927486376542, 13.24441818803487848833252472246, 14.37162041173250348104833287527
