L(s) = 1 | + (−3.74 − 4.24i)2-s + (6.36 + 6.36i)3-s + (−3.95 + 31.7i)4-s + (−41.3 + 41.3i)5-s + (3.15 − 50.8i)6-s − 215. i·7-s + (149. − 102. i)8-s + 81i·9-s + (329. + 20.4i)10-s + (275. − 275. i)11-s + (−227. + 176. i)12-s + (−728. − 728. i)13-s + (−913. + 806. i)14-s − 525.·15-s + (−992. − 251. i)16-s + 1.32e3·17-s + ⋯ |
L(s) = 1 | + (−0.661 − 0.749i)2-s + (0.408 + 0.408i)3-s + (−0.123 + 0.992i)4-s + (−0.739 + 0.739i)5-s + (0.0357 − 0.576i)6-s − 1.66i·7-s + (0.825 − 0.564i)8-s + 0.333i·9-s + (1.04 + 0.0647i)10-s + (0.687 − 0.687i)11-s + (−0.455 + 0.354i)12-s + (−1.19 − 1.19i)13-s + (−1.24 + 1.10i)14-s − 0.603·15-s + (−0.969 − 0.245i)16-s + 1.11·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.358068 - 0.713648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.358068 - 0.713648i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.74 + 4.24i)T \) |
| 3 | \( 1 + (-6.36 - 6.36i)T \) |
good | 5 | \( 1 + (41.3 - 41.3i)T - 3.12e3iT^{2} \) |
| 7 | \( 1 + 215. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-275. + 275. i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (728. + 728. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (1.67e3 + 1.67e3i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 - 1.36e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (4.95e3 + 4.95e3i)T + 2.05e7iT^{2} \) |
| 31 | \( 1 - 6.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-4.05e3 + 4.05e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 4.42e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-968. + 968. i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + 7.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (7.48e3 - 7.48e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (9.19e3 - 9.19e3i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (1.83e4 + 1.83e4i)T + 8.44e8iT^{2} \) |
| 67 | \( 1 + (-2.07e4 - 2.07e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 4.22e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.87e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.80e4 - 1.80e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 2.11e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 8.57e4T + 8.58e9T^{2} \) |
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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
−14.20631635076552841126695314103, −13.03146426858510849652479977070, −11.46647848708855076226461491870, −10.60819300760428293437313905499, −9.720772903661276914725843239297, −7.983970025033820665337853481989, −7.23182398855135002958857267692, −4.14875242014814519568932073112, −3.09804759883997306326988154022, −0.49534405953500056728701276028,
1.86092836330510412142161747618, 4.73822501489064724164700843604, 6.36192640348497474419929117946, 7.83731334506587807822633557153, 8.802123295248987327024390677130, 9.670848473852151214842520381315, 11.90882981870723684721300048371, 12.45145470886697977633481047400, 14.56066400600424022437924299400, 14.90992187717370098448707679554