| L(s) = 1 | + (−5.00 + 4.64i)2-s + (−20.5 + 3.10i)3-s + (1.09 − 14.6i)4-s + (17.9 − 45.6i)5-s + (88.7 − 111. i)6-s + (−123. − 40.1i)7-s + (−73.8 − 92.6i)8-s + (182. − 56.2i)9-s + (122. + 311. i)10-s + (227. + 70.2i)11-s + (22.8 + 304. i)12-s + (212. + 932. i)13-s + (803. − 371. i)14-s + (−227. + 995. i)15-s + (1.26e3 + 190. i)16-s + (−483. + 329. i)17-s + ⋯ |
| L(s) = 1 | + (−0.885 + 0.821i)2-s + (−1.32 + 0.199i)3-s + (0.0342 − 0.456i)4-s + (0.320 − 0.816i)5-s + (1.00 − 1.26i)6-s + (−0.950 − 0.309i)7-s + (−0.407 − 0.511i)8-s + (0.750 − 0.231i)9-s + (0.386 + 0.985i)10-s + (0.567 + 0.174i)11-s + (0.0457 + 0.610i)12-s + (0.349 + 1.53i)13-s + (1.09 − 0.506i)14-s + (−0.260 + 1.14i)15-s + (1.23 + 0.186i)16-s + (−0.406 + 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.465579 + 0.185648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.465579 + 0.185648i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (123. + 40.1i)T \) |
| good | 2 | \( 1 + (5.00 - 4.64i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (20.5 - 3.10i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-17.9 + 45.6i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-227. - 70.2i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (-212. - 932. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (483. - 329. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (1.19e3 + 2.07e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.65e3 - 1.12e3i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-2.94e3 - 1.41e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-4.78e3 + 8.28e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (106. + 1.41e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (-5.99e3 - 7.51e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (3.11e3 - 3.90e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-1.48e4 + 1.37e4i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (1.39e3 - 1.85e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-314. - 800. i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (2.56e3 + 3.41e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-2.36e4 + 4.09e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.44e3 + 1.65e3i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.72e4 + 2.52e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (1.40e3 + 2.43e3i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.22e4 - 9.72e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-1.02e5 + 3.15e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 + 6.05e4T + 8.58e9T^{2} \) |
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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
−15.40367278541536905665105918870, −13.41883007237959667691770532989, −12.33440313723109414278534462113, −11.10854393008756427602950405719, −9.587593765636592543183080669003, −8.875544620823623727787367939675, −6.85180762002908779392709956129, −6.20963253159973933779495037285, −4.45012776735282225762388537976, −0.68404886095587150678514140104,
0.76509772131933021623953776202, 2.90240235148415571203787040209, 5.72240904943360001750016761308, 6.57617405128656812245291036859, 8.659716020360052486747252728341, 10.31389148445359364751261413817, 10.55494194426925062476164850741, 11.87525584211560908227929789227, 12.72709457886850032373062606872, 14.49100155143364101777707645926
