L(s) = 1 | + 0.934·2-s + 1.73·3-s − 3.12·4-s + 0.661i·5-s + 1.61·6-s − 2.64i·7-s − 6.65·8-s + 2.99·9-s + 0.617i·10-s + 14.7i·11-s − 5.41·12-s + 10.6·13-s − 2.47i·14-s + 1.14i·15-s + 6.29·16-s + 15.7i·17-s + ⋯ |
L(s) = 1 | + 0.467·2-s + 0.577·3-s − 0.781·4-s + 0.132i·5-s + 0.269·6-s − 0.377i·7-s − 0.832·8-s + 0.333·9-s + 0.0617i·10-s + 1.33i·11-s − 0.451·12-s + 0.817·13-s − 0.176i·14-s + 0.0763i·15-s + 0.393·16-s + 0.927i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.025685729\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025685729\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (-20.5 - 10.3i)T \) |
good | 2 | \( 1 - 0.934T + 4T^{2} \) |
| 5 | \( 1 - 0.661iT - 25T^{2} \) |
| 11 | \( 1 - 14.7iT - 121T^{2} \) |
| 13 | \( 1 - 10.6T + 169T^{2} \) |
| 17 | \( 1 - 15.7iT - 289T^{2} \) |
| 19 | \( 1 - 14.9iT - 361T^{2} \) |
| 29 | \( 1 + 22.7T + 841T^{2} \) |
| 31 | \( 1 + 1.01T + 961T^{2} \) |
| 37 | \( 1 - 36.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 0.910iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 21.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 20.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 47.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 40.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 70.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 29.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 13.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 61.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 93.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 34.1iT - 9.40e3T^{2} \) |
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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
−10.77186117257386907020885484667, −9.968491552555316416173280010906, −9.127875485871410967193977272349, −8.309131684705469536730547666077, −7.33428460789247681646857511613, −6.23731055704034668380938478793, −5.00409121481149978135193775250, −4.08934565106977240699758636384, −3.23417437336664439617729833265, −1.51929679973589587454126514419,
0.73843247786484363496580704771, 2.81485253329867431353163095966, 3.64436220365908600516749195014, 4.86372750127420255588497633042, 5.70366324591853593946474636389, 6.85236931124116265698107753087, 8.209908363685896555587602341686, 8.906872146906250694268468209332, 9.322554948369854596389406800534, 10.72511019505678835029966635646