| L(s) = 1 | − i·2-s − 4-s − 2i·7-s + i·8-s + 3·11-s − 4i·13-s − 2·14-s + 16-s − 3i·17-s − 5·19-s − 3i·22-s − 6i·23-s − 4·26-s + 2i·28-s + 2·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s + 0.904·11-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 0.727i·17-s − 1.14·19-s − 0.639i·22-s − 1.25i·23-s − 0.784·26-s + 0.377i·28-s + 0.359·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.649403 - 1.05075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.649403 - 1.05075i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71432999558965687424480725120, −10.15627762693614506314734218646, −9.101289914422487788296466710397, −8.234476345716487980298940491831, −7.13740507175189545308739624810, −6.06601673927401130057705833198, −4.70701752447342561505061485988, −3.82711533611488512591428116161, −2.54363682198174938006208109249, −0.819192104519463078705639082930,
1.87207435922589792516046536963, 3.68938504086695471403930522546, 4.70479388462989093193336425987, 5.98474404169795160238665735821, 6.56959845574219540452047617120, 7.69602915479041042990336412423, 8.815797402575775945540480326978, 9.203423758695799293207511199080, 10.36488722718985008200889711497, 11.55559769687808258146981381899
