| L(s) = 1 | + (−0.105 − 0.289i)2-s + (−1.61 − 0.285i)3-s + (1.45 − 1.22i)4-s + (0.0879 + 0.498i)6-s + (−0.0772 + 0.0445i)7-s + (−1.04 − 0.601i)8-s + (−0.279 − 0.101i)9-s + (1.68 − 2.91i)11-s + (−2.71 + 1.56i)12-s + (−0.209 + 0.0369i)13-s + (0.0210 + 0.0176i)14-s + (0.597 − 3.38i)16-s + (−0.859 − 2.36i)17-s + 0.0916i·18-s + (−0.949 + 4.25i)19-s + ⋯ |
| L(s) = 1 | + (−0.0744 − 0.204i)2-s + (−0.934 − 0.164i)3-s + (0.729 − 0.612i)4-s + (0.0358 + 0.203i)6-s + (−0.0291 + 0.0168i)7-s + (−0.368 − 0.212i)8-s + (−0.0931 − 0.0339i)9-s + (0.507 − 0.879i)11-s + (−0.782 + 0.452i)12-s + (−0.0581 + 0.0102i)13-s + (0.00562 + 0.00471i)14-s + (0.149 − 0.846i)16-s + (−0.208 − 0.573i)17-s + 0.0215i·18-s + (−0.217 + 0.975i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.341304 - 0.799893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.341304 - 0.799893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (0.949 - 4.25i)T \) |
| good | 2 | \( 1 + (0.105 + 0.289i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (1.61 + 0.285i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (0.0772 - 0.0445i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.68 + 2.91i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.209 - 0.0369i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.859 + 2.36i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (3.83 + 4.57i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (4.51 + 1.64i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.03 + 6.98i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.84iT - 37T^{2} \) |
| 41 | \( 1 + (0.523 - 2.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.57 + 1.87i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.60 + 7.15i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (5.39 + 6.43i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-9.80 + 3.56i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.757 - 0.635i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.41 - 9.37i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.73 - 3.97i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-15.5 - 2.73i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.178 + 1.01i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-15.5 + 8.96i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.113 + 0.646i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.75 - 15.7i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.89814685070230074710489677310, −10.02265481867666492344114546623, −9.051391822650229738577946470491, −7.84604426790687502664133864436, −6.61644250010233084178112340414, −6.05958010381400945891675511304, −5.29536582837465421705310018489, −3.70961278115624655335074802403, −2.19287237846408890556417884896, −0.57916816143042484901875102599,
1.98591001898160971559047580554, 3.50310395039775829149424521419, 4.77148560742330699547967079481, 5.87515690843097660088185887537, 6.71244050379356448203043719445, 7.49177440836574040229888428805, 8.582325784841249972349126221096, 9.597395280492646696832280341996, 10.81143440946986212956251372600, 11.22434588510937549722139154580
