L(s) = 1 | − 9.03i·5-s − 2.64i·7-s + 12.4·11-s − 9.03i·13-s − 12.3·17-s − 28.8·19-s − 24.6i·23-s − 56.5·25-s − 22.4i·29-s + 16.7i·31-s − 23.8·35-s + 16.2i·37-s − 6.97·41-s + 22.8·43-s − 6.19i·47-s + ⋯ |
L(s) = 1 | − 1.80i·5-s − 0.377i·7-s + 1.13·11-s − 0.694i·13-s − 0.726·17-s − 1.51·19-s − 1.07i·23-s − 2.26·25-s − 0.774i·29-s + 0.541i·31-s − 0.682·35-s + 0.439i·37-s − 0.170·41-s + 0.530·43-s − 0.131i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.095448786\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.095448786\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 5 | \( 1 + 9.03iT - 25T^{2} \) |
| 11 | \( 1 - 12.4T + 121T^{2} \) |
| 13 | \( 1 + 9.03iT - 169T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 + 28.8T + 361T^{2} \) |
| 23 | \( 1 + 24.6iT - 529T^{2} \) |
| 29 | \( 1 + 22.4iT - 841T^{2} \) |
| 31 | \( 1 - 16.7iT - 961T^{2} \) |
| 37 | \( 1 - 16.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6.97T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 6.19iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.01iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 30.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 15.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 78.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 40.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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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
−8.471030005424546023675429740899, −8.153181550546716505716359419307, −6.84652167615591906740433366345, −6.16351660299035230804707268157, −5.16023828800970812742764052520, −4.38585497325716329249657817517, −3.90137888536371055315970239886, −2.26741916803716910060841563111, −1.18077856323363786701161796721, −0.28273704292989391192593916218,
1.79278876231269490385725203916, 2.55967722697070426605631371167, 3.64162190836668500940307010249, 4.24926564155713104994021166025, 5.66896901177805412886437250745, 6.57065785766242096724336441415, 6.75714806094803416210419855336, 7.68224299325032650127223719851, 8.745596539678560081962928010114, 9.390869001193260720039625867867