| L(s) = 1 | + (−0.994 − 0.505i)2-s + (2.83 − 2.68i)3-s + (−1.62 − 2.22i)4-s + (2.17 − 0.0232i)5-s + (−4.17 + 1.23i)6-s + (6.40 − 2.83i)7-s + (1.20 + 7.43i)8-s + (0.334 − 6.24i)9-s + (−2.17 − 1.07i)10-s + (12.8 − 11.4i)11-s + (−10.5 − 1.94i)12-s + (5.09 + 3.09i)13-s + (−7.79 − 0.419i)14-s + (6.10 − 5.91i)15-s + (−0.800 + 2.50i)16-s + (3.26 − 0.386i)17-s + ⋯ |
| L(s) = 1 | + (−0.497 − 0.252i)2-s + (0.944 − 0.895i)3-s + (−0.406 − 0.556i)4-s + (0.435 − 0.00465i)5-s + (−0.695 + 0.206i)6-s + (0.914 − 0.404i)7-s + (0.150 + 0.929i)8-s + (0.0371 − 0.694i)9-s + (−0.217 − 0.107i)10-s + (1.17 − 1.04i)11-s + (−0.882 − 0.162i)12-s + (0.392 + 0.237i)13-s + (−0.556 − 0.0299i)14-s + (0.407 − 0.394i)15-s + (−0.0500 + 0.156i)16-s + (0.192 − 0.0227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13363 - 1.60306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13363 - 1.60306i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-6.40 + 2.83i)T \) |
| good | 2 | \( 1 + (0.994 + 0.505i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-2.83 + 2.68i)T + (0.480 - 8.98i)T^{2} \) |
| 5 | \( 1 + (-2.17 + 0.0232i)T + (24.9 - 0.534i)T^{2} \) |
| 11 | \( 1 + (-12.8 + 11.4i)T + (14.1 - 120. i)T^{2} \) |
| 13 | \( 1 + (-5.09 - 3.09i)T + (78.1 + 149. i)T^{2} \) |
| 17 | \( 1 + (-3.26 + 0.386i)T + (281. - 67.3i)T^{2} \) |
| 19 | \( 1 + (-15.6 + 9.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.44 + 4.18i)T + (-417. + 325. i)T^{2} \) |
| 29 | \( 1 + (44.0 - 8.58i)T + (779. - 315. i)T^{2} \) |
| 31 | \( 1 + (24.1 - 9.48i)T + (704. - 653. i)T^{2} \) |
| 37 | \( 1 + (-17.8 - 7.67i)T + (941. + 993. i)T^{2} \) |
| 41 | \( 1 + (55.9 - 39.0i)T + (580. - 1.57e3i)T^{2} \) |
| 43 | \( 1 + (8.39 - 51.9i)T + (-1.75e3 - 582. i)T^{2} \) |
| 47 | \( 1 + (15.4 + 3.88i)T + (1.94e3 + 1.04e3i)T^{2} \) |
| 53 | \( 1 + (-21.3 + 77.9i)T + (-2.41e3 - 1.43e3i)T^{2} \) |
| 59 | \( 1 + (16.4 - 7.70i)T + (2.22e3 - 2.67e3i)T^{2} \) |
| 61 | \( 1 + (-0.799 - 18.6i)T + (-3.70e3 + 317. i)T^{2} \) |
| 67 | \( 1 + (-90.3 + 13.6i)T + (4.28e3 - 1.32e3i)T^{2} \) |
| 71 | \( 1 + (-5.16 - 1.00i)T + (4.67e3 + 1.89e3i)T^{2} \) |
| 73 | \( 1 + (118. - 29.8i)T + (4.69e3 - 2.51e3i)T^{2} \) |
| 79 | \( 1 + (6.67 + 89.1i)T + (-6.17e3 + 930. i)T^{2} \) |
| 83 | \( 1 + (-91.0 + 8.77i)T + (6.76e3 - 1.31e3i)T^{2} \) |
| 89 | \( 1 + (75.8 - 31.6i)T + (5.57e3 - 5.63e3i)T^{2} \) |
| 97 | \( 1 + (-43.7 - 34.8i)T + (2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.15343449201667262370205802601, −9.861759253728273626814589975012, −9.006387651778390231822858011695, −8.394150720912827720197976369483, −7.49371800615343798766284688219, −6.27247065591180262361123282989, −5.10436987517524735133342346438, −3.55808439527686573896619196718, −1.87499959771667473448253413528, −1.14464093649388680831236028999,
1.83498417242839549466039171603, 3.56054640909138737410813687898, 4.25424911543760954499770618139, 5.58193694148686724519919868033, 7.20716997023590652043373830191, 8.044312324958757420733404612339, 8.995721027308417308673659761682, 9.430849322218411531786280088461, 10.18238145385944119947095335571, 11.59329519007145265018850259046
