L(s) = 1 | + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (2.08 − 4.54i)5-s − 2.44·6-s + (−1.45 − 1.45i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.62 + 2.46i)10-s + 6.56·11-s + (2.44 + 2.44i)12-s + (16.0 − 16.0i)13-s + 2.91i·14-s + (−3.01 − 8.11i)15-s − 4·16-s + (10.8 + 10.8i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.416 − 0.909i)5-s − 0.408·6-s + (−0.208 − 0.208i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.662 + 0.246i)10-s + 0.597·11-s + (0.204 + 0.204i)12-s + (1.23 − 1.23i)13-s + 0.208i·14-s + (−0.201 − 0.541i)15-s − 0.250·16-s + (0.639 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.808597528\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808597528\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-2.08 + 4.54i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (1.45 + 1.45i)T + 49iT^{2} \) |
| 11 | \( 1 - 6.56T + 121T^{2} \) |
| 13 | \( 1 + (-16.0 + 16.0i)T - 169iT^{2} \) |
| 17 | \( 1 + (-10.8 - 10.8i)T + 289iT^{2} \) |
| 19 | \( 1 - 8.18iT - 361T^{2} \) |
| 29 | \( 1 + 6.33iT - 841T^{2} \) |
| 31 | \( 1 - 27.7T + 961T^{2} \) |
| 37 | \( 1 + (14.1 + 14.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 69.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-17.6 + 17.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-22.1 - 22.1i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (1.75 - 1.75i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 84.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 30.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + (30.7 + 30.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 41.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (37.1 - 37.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 95.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-44.3 + 44.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 142. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-99.3 - 99.3i)T + 9.40e3iT^{2} \) |
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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
−10.03690500810855675693517745648, −8.919620679943184941921818982396, −8.458015707902574403130614661753, −7.67511552197452232519932649797, −6.39777786820299378252314867437, −5.55071789458825493773133734957, −4.08011318150338156249149140135, −3.17298355002318166231187587411, −1.66904039533804646922133946664, −0.800936009952168107189647594589,
1.53685415611106890549383814598, 2.91547465443564677307478543502, 3.99402726152262665989570913040, 5.31703803326378338398035583587, 6.45493264991389391924245250906, 6.87184667424617427933091748077, 8.066910769996871566089673130106, 9.014390048861963118195639984963, 9.527084453589794467436465564925, 10.37048682060920323416371664953