L(s) = 1 | − i·2-s − i·3-s − 4-s + (2.21 + 0.311i)5-s − 6-s − 0.622i·7-s + i·8-s − 9-s + (0.311 − 2.21i)10-s + 4.42·11-s + i·12-s + 1.37i·13-s − 0.622·14-s + (0.311 − 2.21i)15-s + 16-s − 6.42i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.990 + 0.139i)5-s − 0.408·6-s − 0.235i·7-s + 0.353i·8-s − 0.333·9-s + (0.0983 − 0.700i)10-s + 1.33·11-s + 0.288i·12-s + 0.382i·13-s − 0.166·14-s + (0.0803 − 0.571i)15-s + 0.250·16-s − 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16191 - 1.33657i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16191 - 1.33657i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.21 - 0.311i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 0.622iT - 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 1.37iT - 13T^{2} \) |
| 17 | \( 1 + 6.42iT - 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.05iT - 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 3.37iT - 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 - 2.75iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.19iT - 83T^{2} \) |
| 89 | \( 1 + 2.75T + 89T^{2} \) |
| 97 | \( 1 - 2.13iT - 97T^{2} \) |
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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
−10.23419076093685965307146247809, −9.249311416817895034016428195055, −9.007347732895186106275361086333, −7.47196905249769108096788146089, −6.72178235877510302429754953997, −5.75226381288091066140001981433, −4.67200353072705918438445387483, −3.36444271991513187527168419979, −2.20330752512858384749545525974, −1.10442929055303463146516145853,
1.57560809319713457757397663122, 3.31249983328288677068321483242, 4.42056191506382165798247338144, 5.44541610131936594297340118509, 6.19631737320808570744148549817, 6.92979940637423393874711620708, 8.531516920192260308199570271724, 8.713377022360766856081001092503, 9.951442228148650106749997038885, 10.22371823113874228432298043510