| L(s) = 1 | + (10.2 + 17.7i)2-s + (953. + 550. i)3-s + (1.83e3 − 3.18e3i)4-s + (−7.38e3 + 4.26e3i)5-s + 2.25e4i·6-s + (1.06e5 + 4.92e4i)7-s + 1.59e5·8-s + (3.39e5 + 5.88e5i)9-s + (−1.51e5 − 8.73e4i)10-s + (−8.08e5 + 1.40e6i)11-s + (3.50e6 − 2.02e6i)12-s − 1.05e6i·13-s + (2.21e5 + 2.40e6i)14-s − 9.38e6·15-s + (−5.89e6 − 1.02e7i)16-s + (−3.25e7 − 1.88e7i)17-s + ⋯ |
| L(s) = 1 | + (0.160 + 0.277i)2-s + (1.30 + 0.754i)3-s + (0.448 − 0.777i)4-s + (−0.472 + 0.272i)5-s + 0.483i·6-s + (0.908 + 0.418i)7-s + 0.607·8-s + (0.639 + 1.10i)9-s + (−0.151 − 0.0873i)10-s + (−0.456 + 0.790i)11-s + (1.17 − 0.677i)12-s − 0.218i·13-s + (0.0293 + 0.318i)14-s − 0.823·15-s + (−0.351 − 0.608i)16-s + (−1.34 − 0.778i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(2.52536 + 0.982667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52536 + 0.982667i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.06e5 - 4.92e4i)T \) |
| good | 2 | \( 1 + (-10.2 - 17.7i)T + (-2.04e3 + 3.54e3i)T^{2} \) |
| 3 | \( 1 + (-953. - 550. i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (7.38e3 - 4.26e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (8.08e5 - 1.40e6i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 + 1.05e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (3.25e7 + 1.88e7i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-6.31e7 + 3.64e7i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (7.54e7 + 1.30e8i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 + 8.76e7T + 3.53e17T^{2} \) |
| 31 | \( 1 + (9.82e8 + 5.67e8i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-1.53e9 - 2.65e9i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 + 1.94e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 2.80e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (4.52e9 - 2.60e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (2.08e10 - 3.61e10i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-2.90e9 - 1.67e9i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (1.62e9 - 9.35e8i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-4.93e10 + 8.54e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 - 1.90e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (1.09e11 + 6.33e10i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-9.80e8 - 1.69e9i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 - 6.01e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (4.54e10 - 2.62e10i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 - 8.01e10iT - 6.93e23T^{2} \) |
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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
−20.02481720502384926422993589351, −18.33809863213561818365786812945, −15.65292424051136710979595094035, −15.13331143596933795805972798899, −13.92831440106971808593822468162, −11.15109564259020099263611200725, −9.427717959918794822973705355182, −7.58445148914738169809800702980, −4.75093511570035561174083627152, −2.39331667261815248629894770603,
1.86325291258429835444419567438, 3.67438105242148855301108061982, 7.54343349331045686795785076120, 8.412059302460858251601358579310, 11.36820093252241862571429642182, 12.99712679535799177317937486508, 14.17320920376662374128112921806, 16.02858065365978834493543025673, 17.94180861667953237018175449252, 19.67083985115092148163880809397
