| L(s) = 1 | + 4.49i·3-s + 8.74·5-s − 7.73i·7-s − 11.1·9-s − 20.6i·11-s + 4.10·13-s + 39.2i·15-s − 12.3·17-s + 4.35i·19-s + 34.7·21-s + 0.917i·23-s + 51.5·25-s − 9.71i·27-s − 40.2·29-s − 37.8i·31-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s + 1.74·5-s − 1.10i·7-s − 1.24·9-s − 1.87i·11-s + 0.316·13-s + 2.61i·15-s − 0.726·17-s + 0.229i·19-s + 1.65·21-s + 0.0398i·23-s + 2.06·25-s − 0.359i·27-s − 1.38·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.627482858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627482858\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 4.49iT - 9T^{2} \) |
| 5 | \( 1 - 8.74T + 25T^{2} \) |
| 7 | \( 1 + 7.73iT - 49T^{2} \) |
| 11 | \( 1 + 20.6iT - 121T^{2} \) |
| 13 | \( 1 - 4.10T + 169T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 23 | \( 1 - 0.917iT - 529T^{2} \) |
| 29 | \( 1 + 40.2T + 841T^{2} \) |
| 31 | \( 1 + 37.8iT - 961T^{2} \) |
| 37 | \( 1 - 47.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.91iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 9.69T + 3.72e3T^{2} \) |
| 67 | \( 1 - 7.57iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 113.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 100. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 24.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 108.T + 9.40e3T^{2} \) |
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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
−9.583016146852391135924582963968, −9.091024730121276752615261989952, −8.213755934620881377390248958596, −6.83633359733152854798002257947, −5.78881523204582529945167316135, −5.52744820942617996043481496789, −4.22278984467302843859486317568, −3.57285256544568158381676864766, −2.36497400907795238213974056682, −0.78532194566550536430513791130,
1.38321854229156541344200678531, 2.12777896644640530899391161005, 2.57395368307686151570983316967, 4.66882769156511525996921928591, 5.63894379106106097495041574487, 6.25502179884028064137434228438, 6.93361325058374819338260568357, 7.71576775214361121623923321321, 8.928832352039284951507100134193, 9.332379157961176898051271373604
