| L(s) = 1 | + 2.82·5-s + 26·11-s − 33.9·13-s − 103.·17-s − 94.7·19-s + 148·23-s − 117·25-s + 118·29-s + 296.·31-s − 254·37-s + 91.9·41-s + 122·43-s + 308.·47-s + 170·53-s + 73.5·55-s + 304.·59-s + 608.·61-s − 96·65-s + 420·67-s − 420·71-s − 813.·73-s + 1.05e3·79-s − 1.44e3·83-s − 292·85-s − 1.02e3·89-s − 268·95-s − 315.·97-s + ⋯ |
| L(s) = 1 | + 0.252·5-s + 0.712·11-s − 0.724·13-s − 1.47·17-s − 1.14·19-s + 1.34·23-s − 0.936·25-s + 0.755·29-s + 1.72·31-s − 1.12·37-s + 0.350·41-s + 0.432·43-s + 0.956·47-s + 0.440·53-s + 0.180·55-s + 0.670·59-s + 1.27·61-s − 0.183·65-s + 0.765·67-s − 0.702·71-s − 1.30·73-s + 1.49·79-s − 1.91·83-s − 0.372·85-s − 1.22·89-s − 0.289·95-s − 0.330·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.988020233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988020233\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 125T^{2} \) |
| 11 | \( 1 - 26T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148T + 1.21e4T^{2} \) |
| 29 | \( 1 - 118T + 2.43e4T^{2} \) |
| 31 | \( 1 - 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122T + 7.95e4T^{2} \) |
| 47 | \( 1 - 308.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 170T + 1.48e5T^{2} \) |
| 59 | \( 1 - 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 608.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 + 813.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 315.T + 9.12e5T^{2} \) |
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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
−8.830842881783849142766348003254, −8.396243361388816267020363083481, −7.10649665972536444228241389120, −6.68175554038233748465946261558, −5.77920910852717602051301480721, −4.68535050791154122464821176388, −4.12138979847214146905674278589, −2.78627803163962599803158721574, −1.97994653624427232276554070878, −0.65547361154197423744751226787,
0.65547361154197423744751226787, 1.97994653624427232276554070878, 2.78627803163962599803158721574, 4.12138979847214146905674278589, 4.68535050791154122464821176388, 5.77920910852717602051301480721, 6.68175554038233748465946261558, 7.10649665972536444228241389120, 8.396243361388816267020363083481, 8.830842881783849142766348003254
