L(s) = 1 | + (−2.41 − 1.47i)2-s + 4.79·3-s + (3.65 + 7.11i)4-s − 17.0i·5-s + (−11.5 − 7.07i)6-s + (18.3 − 2.33i)7-s + (1.65 − 22.5i)8-s − 3.97·9-s + (−25.1 + 41.2i)10-s + 41.4i·11-s + (17.5 + 34.1i)12-s + 45.3i·13-s + (−47.7 − 21.4i)14-s − 81.9i·15-s + (−37.2 + 52.0i)16-s − 28.2i·17-s + ⋯ |
L(s) = 1 | + (−0.853 − 0.521i)2-s + 0.923·3-s + (0.457 + 0.889i)4-s − 1.52i·5-s + (−0.788 − 0.481i)6-s + (0.992 − 0.126i)7-s + (0.0732 − 0.997i)8-s − 0.147·9-s + (−0.795 + 1.30i)10-s + 1.13i·11-s + (0.422 + 0.821i)12-s + 0.967i·13-s + (−0.912 − 0.409i)14-s − 1.41i·15-s + (−0.582 + 0.813i)16-s − 0.403i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.960593 - 0.505934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.960593 - 0.505934i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.41 + 1.47i)T \) |
| 7 | \( 1 + (-18.3 + 2.33i)T \) |
good | 3 | \( 1 - 4.79T + 27T^{2} \) |
| 5 | \( 1 + 17.0iT - 125T^{2} \) |
| 11 | \( 1 - 41.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 28.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 41.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 81.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 171. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 575.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 778. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 198. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 197. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 255. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 938.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 656. iT - 9.12e5T^{2} \) |
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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
−16.89738113654140841953818268406, −15.50932352435908006359974853742, −13.96237853127009830601449595016, −12.59841873543573690361887382827, −11.49399383120051440443400144331, −9.542361998127801261337529105601, −8.713621286185369342635717612134, −7.66656253026416695120763702419, −4.46880084781705925586991738794, −1.80429656819050611382868374484,
2.74031707088156760222472367646, 6.08054719674180413494880607669, 7.75966662624519035024875089350, 8.616673585881805295965825832806, 10.43564344324314236145253542904, 11.24714635778300143825514149393, 13.94147825199111461388058338528, 14.65465888818022407623206033554, 15.38348077779876654972407563875, 17.14104383010778718829746472696