L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−1 + 3.07i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (3.04 − 1.31i)11-s − 12-s + (−2.73 + 1.98i)13-s + (−1 − 3.07i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (4.92 + 3.57i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.330 + 0.239i)6-s + (−0.377 + 1.16i)7-s + (0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.316·10-s + (0.918 − 0.396i)11-s − 0.288·12-s + (−0.758 + 0.551i)13-s + (−0.267 − 0.822i)14-s + (0.0797 − 0.245i)15-s + (−0.202 − 0.146i)16-s + (1.19 + 0.868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.841158 + 0.466108i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.841158 + 0.466108i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-3.04 + 1.31i)T \) |
good | 7 | \( 1 + (1 - 3.07i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (2.73 - 1.98i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.92 - 3.57i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.381 - 1.17i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 + (0.809 - 2.48i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.11 + 0.812i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.97 - 6.06i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.381 - 1.17i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.0901T + 43T^{2} \) |
| 47 | \( 1 + (1.33 + 4.11i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.85 - 4.97i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.35 + 13.4i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + (8.47 + 6.15i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.14 + 12.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.30 + 2.40i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.23 + 2.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + (15.0 - 10.9i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.84966378252966298335183516819, −10.74876599119282854228473690453, −9.611459607682684373484301817151, −8.983650284425744557287640523569, −7.965956772469849319884819388251, −6.79453812073690855823578665174, −6.13282115157734824177470152705, −5.16966600708629897219699352910, −3.11140913679778807329830191461, −1.61807835429464512134003720748,
0.936447767184423893887300194954, 2.98300796374035956158616815583, 4.17706286804486207038170550852, 5.33554577798137018527696910163, 6.85790833163126549819217649238, 7.58006281429738948417772431548, 9.007096783082925707193462122342, 9.731965985477299149183196941290, 10.28473881986546970609345228030, 11.26603363051978341221040046737