| L(s) = 1 | + 2.78·2-s − 5.54i·3-s + 3.74·4-s − 15.4i·6-s − 4.48·7-s − 0.713·8-s − 21.7·9-s + (2.46 − 10.7i)11-s − 20.7i·12-s + 15.7·13-s − 12.4·14-s − 16.9·16-s + 25.4·17-s − 60.5·18-s − 8.44i·19-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.84i·3-s + 0.935·4-s − 2.57i·6-s − 0.640·7-s − 0.0892·8-s − 2.41·9-s + (0.223 − 0.974i)11-s − 1.73i·12-s + 1.20·13-s − 0.890·14-s − 1.06·16-s + 1.49·17-s − 3.36·18-s − 0.444i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20016 - 2.54504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20016 - 2.54504i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (-2.46 + 10.7i)T \) |
| good | 2 | \( 1 - 2.78T + 4T^{2} \) |
| 3 | \( 1 + 5.54iT - 9T^{2} \) |
| 7 | \( 1 + 4.48T + 49T^{2} \) |
| 13 | \( 1 - 15.7T + 169T^{2} \) |
| 17 | \( 1 - 25.4T + 289T^{2} \) |
| 19 | \( 1 + 8.44iT - 361T^{2} \) |
| 23 | \( 1 - 19.8iT - 529T^{2} \) |
| 29 | \( 1 + 21.4iT - 841T^{2} \) |
| 31 | \( 1 - 24.9T + 961T^{2} \) |
| 37 | \( 1 + 30.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 25.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 93.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 27.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 81.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 49.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 44.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 29.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 47.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 87.8iT - 9.40e3T^{2} \) |
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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
−11.83762442025879502383695474137, −11.02585256397426917861115674405, −9.157925811541106575377986714038, −8.106305577598000537826720450678, −7.04984198507148919849984047141, −5.97589406230501723338242496615, −5.76757882041934181205171187285, −3.66768667983659075430363186327, −2.74428734513402164698945386347, −0.967689605886393849256902239653,
3.04218419561776726297053058748, 3.78125382666609490235782445821, 4.64095135792134662210532963104, 5.58105091074816936269158588780, 6.47279266047505110825797758780, 8.377396471807473322722180359178, 9.485533493346882584852349581405, 10.13219833431919671888447712628, 11.12347148459536922530786863792, 12.08613430934737083462129895814
