| L(s) = 1 | + (1 − 2i)5-s − 2.82i·7-s − 3·9-s + (2.82 − 1.73i)11-s − 4.89·13-s − 4.89·17-s − 5.65·19-s + 6.92·23-s + (−3 − 4i)25-s + 9.79i·29-s + 3.46i·31-s + (−5.65 − 2.82i)35-s − 4i·37-s − 2.82i·43-s + (−3 + 6i)45-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.894i)5-s − 1.06i·7-s − 9-s + (0.852 − 0.522i)11-s − 1.35·13-s − 1.18·17-s − 1.29·19-s + 1.44·23-s + (−0.600 − 0.800i)25-s + 1.81i·29-s + 0.622i·31-s + (−0.956 − 0.478i)35-s − 0.657i·37-s − 0.431i·43-s + (−0.447 + 0.894i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.288005 - 0.915744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.288005 - 0.915744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 11 | \( 1 + (-2.82 + 1.73i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 9.79iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.659516800499834925037939429134, −8.890542515831839164857210479312, −8.401594588318428476821016782500, −7.06865254992068916200368853344, −6.47368797186356385000092027602, −5.19587074054988432012597652504, −4.58003350588003883052568879129, −3.36810739837705713829068728238, −1.95503351903816325788447264384, −0.42527643924973093775334458173,
2.30157554079033013400386036469, 2.64080654140844577428750043608, 4.24001421774667744274861609951, 5.32047242902409071527268527548, 6.30240944239101618778563848067, 6.80830082236058969182363250148, 7.995324441730255351971006892236, 9.016530187025384791652112209438, 9.477404060157148947297343877602, 10.47452085139232499644497790711
