| L(s) = 1 | + (2.12 − 1.44i)2-s + (−0.595 + 0.552i)3-s + (1.68 − 4.29i)4-s + (2.35 + 0.727i)5-s + (−0.465 + 2.03i)6-s + (−1.49 − 6.55i)8-s + (−0.174 + 2.33i)9-s + (6.06 − 1.87i)10-s + (−0.0412 − 0.550i)11-s + (1.37 + 3.49i)12-s + (−1.06 − 0.512i)13-s + (−1.80 + 0.870i)15-s + (−5.91 − 5.48i)16-s + (−7.42 − 1.11i)17-s + (3.00 + 5.21i)18-s + (0.499 − 0.865i)19-s + ⋯ |
| L(s) = 1 | + (1.50 − 1.02i)2-s + (−0.343 + 0.319i)3-s + (0.842 − 2.14i)4-s + (1.05 + 0.325i)5-s + (−0.189 + 0.831i)6-s + (−0.528 − 2.31i)8-s + (−0.0582 + 0.777i)9-s + (1.91 − 0.591i)10-s + (−0.0124 − 0.165i)11-s + (0.395 + 1.00i)12-s + (−0.295 − 0.142i)13-s + (−0.466 + 0.224i)15-s + (−1.47 − 1.37i)16-s + (−1.80 − 0.271i)17-s + (0.709 + 1.22i)18-s + (0.114 − 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.41728 - 1.59327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.41728 - 1.59327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-2.12 + 1.44i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (0.595 - 0.552i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.35 - 0.727i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0412 + 0.550i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (1.06 + 0.512i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (7.42 + 1.11i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.499 + 0.865i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.25 + 0.339i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (3.55 - 4.45i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.90 - 5.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.34 + 3.42i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.206 - 0.902i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.949 - 4.15i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-7.96 + 5.43i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (2.85 - 7.28i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.65 + 0.818i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.60 - 6.63i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (1.62 + 2.81i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0286 - 0.0359i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (6.90 + 4.71i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 5.15i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.337 + 4.50i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 1.37T + 97T^{2} \) |
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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
−11.23997647933398230271775557159, −10.76133844898516972163710191013, −10.05660294919596314717712573093, −8.959106426581183283330593323075, −7.01227615064235244024157553766, −5.99395457998207736890826553674, −5.17478839757024316888197294659, −4.38053391361257257973420631611, −2.85321932987867231711673338867, −1.95863238504783633762570332799,
2.29038635504123471388226693020, 3.89981331679155767186451543105, 4.95039684437514277873022700276, 5.94561921034319747253134765852, 6.47717111541548223080160354907, 7.38310664460491257373038147760, 8.710131283329141166955550974422, 9.690997255935960109640103503536, 11.24183170619619930939787919143, 12.04950931540362558826140204821
