L(s) = 1 | + 2.23·3-s + (1.61 − 1.17i)4-s + (−0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + 2.00·9-s + (1.11 + 0.812i)11-s + (3.61 − 2.62i)12-s + (0.545 + 1.67i)13-s + (−1.80 + 1.31i)15-s + (1.23 − 3.80i)16-s + (−4.11 − 2.99i)17-s + (−0.454 + 1.40i)19-s + (−0.618 + 1.90i)20-s + (−0.690 + 2.12i)21-s + (−2.11 − 6.51i)23-s + ⋯ |
L(s) = 1 | + 1.29·3-s + (0.809 − 0.587i)4-s + (−0.361 + 0.262i)5-s + (−0.116 + 0.359i)7-s + 0.666·9-s + (0.337 + 0.244i)11-s + (1.04 − 0.758i)12-s + (0.151 + 0.465i)13-s + (−0.467 + 0.339i)15-s + (0.309 − 0.951i)16-s + (−0.998 − 0.725i)17-s + (−0.104 + 0.321i)19-s + (−0.138 + 0.425i)20-s + (−0.150 + 0.464i)21-s + (−0.441 − 1.35i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01834 - 0.131724i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01834 - 0.131724i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (3.30 - 5.48i)T \) |
good | 2 | \( 1 + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + (0.809 - 0.587i)T + (1.54 - 4.75i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 0.812i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.545 - 1.67i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (4.11 + 2.99i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.454 - 1.40i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.11 + 6.51i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.67i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.690 + 0.502i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-9.28 + 6.74i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-2.54 - 7.83i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (0.736 + 2.26i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.23 - 3.80i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.59 - 4.89i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.881 - 2.71i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.47 + 4.70i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (6.61 + 4.80i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (3.33 - 10.2i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.66 + 2.66i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.62485657019081161898818624502, −10.95810809091937125418766908284, −9.685608246667631485577814736890, −9.047179766247595012214834087387, −7.925754324728750825277420796620, −7.01560060896927248906610564447, −6.03140978654089467330493000953, −4.36810882369573082562899974044, −2.97586025273578933713172011842, −2.02227895550259129114930214339,
2.05775835243454030106773562796, 3.31755718748994953505151448806, 4.11930577537592518722367611845, 6.08561181962945822486297601784, 7.25555153212601329551119596861, 8.090288058065565087465671664152, 8.679966304207036966162357686327, 9.823697000037752126588096070513, 11.02125292874505600175474643556, 11.80150187520291300546714447929