L(s) = 1 | − 1.17·3-s − 2.23i·5-s + 2.64i·7-s − 1.62·9-s + 0.359·11-s − 5.64i·13-s + 2.62i·15-s − 2.03·17-s − 3.10i·21-s − 5.00·25-s + 5.42·27-s + 10.7i·29-s − 0.422·33-s + 5.91·35-s + 6.62i·39-s + ⋯ |
L(s) = 1 | − 0.677·3-s − 0.999i·5-s + 0.999i·7-s − 0.541·9-s + 0.108·11-s − 1.56i·13-s + 0.677i·15-s − 0.492·17-s − 0.677i·21-s − 1.00·25-s + 1.04·27-s + 1.99i·29-s − 0.0735·33-s + 0.999·35-s + 1.06i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4596376619\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4596376619\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 11 | \( 1 - 0.359T + 11T^{2} \) |
| 13 | \( 1 + 5.64iT - 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.7iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.994561302008714846853602828352, −8.637285257364019144569340645944, −7.921685277167609218077318605658, −6.79214568321244172748752521865, −5.81121358856903259652964970675, −5.38899895476033934153175447645, −4.81523851196650237964420666202, −3.47490104137960477615137754542, −2.48722158026240220387370487902, −1.10589186381375166421641249296,
0.19210587454227862787209171121, 1.83915840068172775813795217436, 2.95390311806928971607570804087, 4.06080367629849997175687530690, 4.62534658954236887062585388545, 5.98350152833076953738796502535, 6.40539141730932296941036028520, 7.12948540945695218419323818546, 7.83346140708982945201594647300, 8.883214841301674045355279734853