| L(s) = 1 | − 0.175i·3-s + 3.69i·7-s + 2.96·9-s + 0.648·21-s + 9.14i·23-s − 1.04i·27-s + 10.3·29-s − 12.7·41-s − 2.96i·43-s − 3.95i·47-s − 6.65·49-s + 6.15·61-s + 10.9i·63-s + 8.18i·67-s + 1.60·69-s + ⋯ |
| L(s) = 1 | − 0.101i·3-s + 1.39i·7-s + 0.989·9-s + 0.141·21-s + 1.90i·23-s − 0.201i·27-s + 1.92·29-s − 1.99·41-s − 0.451i·43-s − 0.577i·47-s − 0.951·49-s + 0.787·61-s + 1.38i·63-s + 0.999i·67-s + 0.193·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.859283297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.859283297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.175iT - 3T^{2} \) |
| 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.14iT - 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 2.96iT - 43T^{2} \) |
| 47 | \( 1 + 3.95iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.15T + 61T^{2} \) |
| 67 | \( 1 - 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.66T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.564299672327996903108788758562, −8.038194605984473392687238965889, −7.05126978821782211943391897693, −6.53402666873216640974815423955, −5.51901918797176949030565844084, −5.09082652579830570529980949913, −4.03432603270184418855965705381, −3.12711380461195135348024704427, −2.18069211587366132910647102342, −1.28644976561872794742822177659,
0.57372597763922028045587010865, 1.53061729892563565847636175894, 2.79760983656697653933147634410, 3.78450621781602904589294708336, 4.50626748564145703209401291594, 4.94233053711861396211075222202, 6.43169365744567370031207064480, 6.69563872788178179098668053295, 7.50209891694416844564853100011, 8.204108745552411177847737895346
