| L(s) = 1 | + (0.492 + 2.15i)2-s + (−5.00 − 4.64i)3-s + (24.4 − 11.7i)4-s + (−28.7 + 4.33i)5-s + (7.54 − 13.0i)6-s + (−67.2 − 116. i)7-s + (81.4 + 102. i)8-s + (−14.6 − 195. i)9-s + (−23.4 − 59.8i)10-s + (−38.8 − 18.7i)11-s + (−176. − 54.5i)12-s + (380. − 968. i)13-s + (218. − 202. i)14-s + (163. + 111. i)15-s + (360. − 452. i)16-s + (251. + 37.9i)17-s + ⋯ |
| L(s) = 1 | + (0.0869 + 0.381i)2-s + (−0.320 − 0.297i)3-s + (0.763 − 0.367i)4-s + (−0.513 + 0.0774i)5-s + (0.0855 − 0.148i)6-s + (−0.519 − 0.899i)7-s + (0.450 + 0.564i)8-s + (−0.0604 − 0.806i)9-s + (−0.0742 − 0.189i)10-s + (−0.0968 − 0.0466i)11-s + (−0.354 − 0.109i)12-s + (0.623 − 1.58i)13-s + (0.297 − 0.276i)14-s + (0.188 + 0.128i)15-s + (0.352 − 0.441i)16-s + (0.211 + 0.0318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.12909 - 0.885734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12909 - 0.885734i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.13e4 + 4.26e3i)T \) |
| good | 2 | \( 1 + (-0.492 - 2.15i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (5.00 + 4.64i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (28.7 - 4.33i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (67.2 + 116. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (38.8 + 18.7i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (-380. + 968. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (-251. - 37.9i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (17.3 - 230. i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (114. - 78.2i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (747. - 693. i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (-1.35e3 - 418. i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (5.32e3 - 9.22e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-2.84e3 - 1.24e4i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-2.37e4 + 1.14e4i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (1.08e4 + 2.76e4i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (1.12e4 - 1.40e4i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-2.40e4 + 7.43e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (141. - 1.89e3i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (3.02e3 + 2.05e3i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (-1.20e4 + 3.06e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (-2.29e4 - 3.96e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.81e4 - 5.39e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (-4.69e4 - 4.35e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (1.53e5 + 7.39e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−15.00936367553976939047355095161, −13.55413987923953551241292643218, −12.29413307423553005745222999127, −11.08601856054859973647799064903, −10.07092005643246617224754621933, −7.974119049103097892644147814532, −6.86468957570787148571137242326, −5.72284249047352068784663364453, −3.44670295885612859641407205095, −0.795431362979381669232983283218,
2.25483347575259298535046758080, 4.07452074872985089475926860488, 6.02446473094750312683637352034, 7.53540565213718918673923071901, 9.098007041402028445235057020742, 10.70570946658766359157910691968, 11.63634873456313392188265904019, 12.47110227724943035876057534405, 13.91820796349381754339804914880, 15.77062337737169614464656112283
