L(s) = 1 | + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (−1 − 0.726i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 0.999·10-s + (−2.54 − 2.12i)11-s − 0.999·12-s + (1.73 − 5.34i)13-s + (−1 + 0.726i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (1.57 + 4.84i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (−0.126 − 0.388i)6-s + (−0.377 − 0.274i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s − 0.316·10-s + (−0.767 − 0.641i)11-s − 0.288·12-s + (0.481 − 1.48i)13-s + (−0.267 + 0.194i)14-s + (−0.208 − 0.151i)15-s + (0.0772 + 0.237i)16-s + (0.381 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.640549 - 1.29206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.640549 - 1.29206i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (2.54 + 2.12i)T \) |
good | 7 | \( 1 + (1 + 0.726i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.73 + 5.34i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.57 - 4.84i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.61 + 1.90i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 0.145T + 23T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.224i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.11 - 3.44i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.97 - 5.06i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.61 + 1.90i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (9.16 - 6.65i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.145 - 0.449i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (2.35 + 1.71i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + (-0.472 - 1.45i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 7.88i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.19 + 6.74i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.23 - 3.80i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (3.90 - 12.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.16687259026203794151963031470, −10.45268419736800118357745351767, −9.496217721167916747085135789053, −8.346807098200605864266798744909, −7.79286147304935546480815734430, −6.19170422630343967325151238835, −5.20719914704941253733190087673, −3.70997624221636420694094543977, −2.83195699459143473572801186695, −0.970268837371295582242613531242,
2.51661609414376649462293063011, 3.82082453585937748784114443466, 4.91243239663821299967650822777, 6.12970518136246865667113092510, 7.21812875903029804720616546149, 7.925335912575709165486973319990, 9.274999369752186504933088313339, 9.659731456886951668417080529170, 11.00147434511321305793709598581, 11.96089375489684221339369170821