| L(s) = 1 | + (−2.69 − 0.843i)2-s + (1.45 + 4.47i)3-s + (6.57 + 4.55i)4-s + (−9.98 − 5.03i)5-s + (−0.148 − 13.3i)6-s + 7.23i·7-s + (−13.9 − 17.8i)8-s + (3.91 − 2.84i)9-s + (22.6 + 22.0i)10-s + (20.7 − 28.5i)11-s + (−10.8 + 36.0i)12-s + (17.4 − 12.6i)13-s + (6.10 − 19.5i)14-s + (8.03 − 52.0i)15-s + (22.4 + 59.9i)16-s + (121. + 39.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.954 − 0.298i)2-s + (0.279 + 0.861i)3-s + (0.821 + 0.569i)4-s + (−0.892 − 0.450i)5-s + (−0.0101 − 0.905i)6-s + 0.390i·7-s + (−0.614 − 0.788i)8-s + (0.145 − 0.105i)9-s + (0.717 + 0.696i)10-s + (0.568 − 0.783i)11-s + (−0.260 + 0.867i)12-s + (0.372 − 0.270i)13-s + (0.116 − 0.372i)14-s + (0.138 − 0.895i)15-s + (0.351 + 0.936i)16-s + (1.73 + 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.08395 + 0.302181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08395 + 0.302181i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.69 + 0.843i)T \) |
| 5 | \( 1 + (9.98 + 5.03i)T \) |
| good | 3 | \( 1 + (-1.45 - 4.47i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 7.23iT - 343T^{2} \) |
| 11 | \( 1 + (-20.7 + 28.5i)T + (-411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 12.6i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-121. - 39.4i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (99.4 + 32.3i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (80.6 - 110. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-195. + 63.6i)T + (1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (81.5 - 251. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (193. - 140. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-207. + 150. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-486. + 158. i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-104. - 322. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-22.4 - 30.9i)T + (-6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (44.2 - 60.9i)T + (-7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-0.682 + 2.09i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (121. + 373. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (45.9 - 63.1i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (216. + 667. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (432. - 1.33e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (958. + 696. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-1.45e3 + 472. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.05608079669927128610970591752, −10.85001026736269831475498580828, −10.14328228626197660646850957812, −8.908700589651621796412078346802, −8.529963764429876781770623819169, −7.34594911036972189303393661719, −5.85399302562749377658832924118, −4.06067363170574099596858729203, −3.25199249303008209041858157932, −1.06200767931552179146557324532,
0.871873221457149792214464679367, 2.36082775914203738091985441056, 4.16911053414180982667493861439, 6.18944047629272429939982296619, 7.18089523543122337293046788911, 7.70140323479170957924825603156, 8.626884980519445446105452965870, 9.998427656169136681156824190449, 10.75220694205747379946312957332, 11.99955339221495619388178009552
