L(s) = 1 | − 7.51·5-s + 7.72·7-s + 9.01·11-s − 10.6i·13-s − 31.2·17-s + (8.20 + 17.1i)19-s − 4.85·23-s + 31.4·25-s + 23.2i·29-s − 7.74i·31-s − 58.0·35-s − 21.4i·37-s − 32.7i·41-s − 7.82·43-s + 50.7·47-s + ⋯ |
L(s) = 1 | − 1.50·5-s + 1.10·7-s + 0.819·11-s − 0.818i·13-s − 1.83·17-s + (0.431 + 0.902i)19-s − 0.211·23-s + 1.25·25-s + 0.800i·29-s − 0.249i·31-s − 1.65·35-s − 0.579i·37-s − 0.798i·41-s − 0.182·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7491969583\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7491969583\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-8.20 - 17.1i)T \) |
good | 5 | \( 1 + 7.51T + 25T^{2} \) |
| 7 | \( 1 - 7.72T + 49T^{2} \) |
| 11 | \( 1 - 9.01T + 121T^{2} \) |
| 13 | \( 1 + 10.6iT - 169T^{2} \) |
| 17 | \( 1 + 31.2T + 289T^{2} \) |
| 23 | \( 1 + 4.85T + 529T^{2} \) |
| 29 | \( 1 - 23.2iT - 841T^{2} \) |
| 31 | \( 1 + 7.74iT - 961T^{2} \) |
| 37 | \( 1 + 21.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 7.82T + 1.84e3T^{2} \) |
| 47 | \( 1 - 50.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 86.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.46T + 3.72e3T^{2} \) |
| 67 | \( 1 + 14.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 79.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 61.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 143.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 53.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124. iT - 9.40e3T^{2} \) |
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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
−8.632351000599981157250073786012, −8.173400010145739487453331488688, −7.47105767177418116476585655357, −6.83854491331714627826929744878, −5.74977119471524573370124296868, −4.79219612369201928032549436770, −4.12939009425740469979860321721, −3.49567701325108314973182585090, −2.19233036291522150580167728350, −1.01132575221340791022013073678,
0.20530407607212001102468875543, 1.47767277041475737609745437005, 2.59591698155896926746913014538, 3.87632130500471085606617969180, 4.39409856405281768890099157280, 4.92634714605359832133155676939, 6.32453495619778065687686605850, 7.01621781252144155880753481690, 7.62053716947542980991129724418, 8.550110634830951586212480877511