L(s) = 1 | − 4·11-s + 2i·13-s − 2i·17-s − 8·19-s − 4i·23-s − 6·29-s − 2i·37-s + 6·41-s + 4i·43-s − 12i·47-s + 7·49-s + 6i·53-s + 12·59-s + 14·61-s + 12i·67-s + ⋯ |
L(s) = 1 | − 1.20·11-s + 0.554i·13-s − 0.485i·17-s − 1.83·19-s − 0.834i·23-s − 1.11·29-s − 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 1.75i·47-s + 49-s + 0.824i·53-s + 1.56·59-s + 1.79·61-s + 1.46i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.273452321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.273452321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.052824808533117022898045536928, −7.20014264183481599696242855430, −6.69686101872249739436284130494, −5.80333976473364281720755581957, −5.20960803097847799986686464430, −4.33990479946822647734133775935, −3.77028558857676281113920748309, −2.45678948286317846682402887910, −2.21019137744819749006051435412, −0.63749876663459055244269537054,
0.46617264183957075442671633176, 1.88910809898938296870053387701, 2.54752033829613595268143483912, 3.54408379319472328829457109447, 4.24349008523421862213581816283, 5.15410846111931478084596954858, 5.72655344255255830159392598240, 6.41182579100004697809113587017, 7.29609051134082800358814113612, 7.906287554267644760110990007568