| L(s) = 1 | + (−0.990 + 0.360i)2-s + (−0.680 + 0.571i)4-s + (−0.303 − 1.71i)5-s + (1.88 + 1.58i)7-s + (1.52 − 2.63i)8-s + (0.920 + 1.59i)10-s + (−0.217 + 1.23i)11-s + (4.27 + 1.55i)13-s + (−2.43 − 0.886i)14-s + (−0.249 + 1.41i)16-s + (3.32 + 5.75i)17-s + (−0.124 + 0.215i)19-s + (1.18 + 0.996i)20-s + (−0.229 − 1.30i)22-s + (0.645 − 0.541i)23-s + ⋯ |
| L(s) = 1 | + (−0.700 + 0.254i)2-s + (−0.340 + 0.285i)4-s + (−0.135 − 0.768i)5-s + (0.712 + 0.597i)7-s + (0.538 − 0.932i)8-s + (0.291 + 0.504i)10-s + (−0.0656 + 0.372i)11-s + (1.18 + 0.431i)13-s + (−0.651 − 0.237i)14-s + (−0.0622 + 0.353i)16-s + (0.806 + 1.39i)17-s + (−0.0285 + 0.0495i)19-s + (0.265 + 0.222i)20-s + (−0.0489 − 0.277i)22-s + (0.134 − 0.112i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.807017 + 0.291343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.807017 + 0.291343i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.990 - 0.360i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.303 + 1.71i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 1.58i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.217 - 1.23i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.27 - 1.55i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.32 - 5.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.124 - 0.215i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.645 + 0.541i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.481 + 0.175i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.628 - 0.527i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.66 - 2.78i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.751 + 4.26i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.06 + 3.40i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (0.522 + 2.96i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.20 - 1.85i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.47 + 3.44i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.0447 + 0.0774i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.66 + 4.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.48 - 1.63i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.55 - 2.75i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.35 - 5.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.953 + 5.40i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.39569865402668395077297037512, −11.19335690187309661662082817664, −10.10793577755414554764456187927, −8.931486330220784888380389607517, −8.486458593839579895236801202660, −7.66251619155998029412716632826, −6.19750461512008778997468371218, −4.87742104242916249495025758002, −3.78639640672576398116060315644, −1.44832584923555268739739626040,
1.12416429421508735974967537136, 3.10088530825599935566026805950, 4.65152462682116415671037517756, 5.86350207074732831607139562346, 7.32601760335320778856789348054, 8.124345096633138129073106224441, 9.164425125043598754115614372831, 10.19599462261897405680054983330, 11.00435809822727011569923045972, 11.43133398690623202809009235188
