| L(s) = 1 | + 3.16·5-s + 6.32·11-s − 3.16·17-s − 7·19-s − 3.16·23-s + 5.00·25-s + 3.16·29-s + 3·31-s − 4·37-s + 9.48·41-s + 5·43-s + 9.48·47-s + 9.48·53-s + 20.0·55-s + 12.6·59-s − 3·61-s + 10·67-s − 12.6·71-s − 5·73-s + 12·79-s + 6.32·83-s − 10.0·85-s − 9.48·89-s − 22.1·95-s − 5·97-s − 6.32·101-s + 2·103-s + ⋯ |
| L(s) = 1 | + 1.41·5-s + 1.90·11-s − 0.766·17-s − 1.60·19-s − 0.659·23-s + 1.00·25-s + 0.587·29-s + 0.538·31-s − 0.657·37-s + 1.48·41-s + 0.762·43-s + 1.38·47-s + 1.30·53-s + 2.69·55-s + 1.64·59-s − 0.384·61-s + 1.22·67-s − 1.50·71-s − 0.585·73-s + 1.35·79-s + 0.694·83-s − 1.08·85-s − 1.00·89-s − 2.27·95-s − 0.507·97-s − 0.629·101-s + 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.940816578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.940816578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 5T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.536396632589430445582287448371, −7.27071590728183928212578975815, −6.48570110285700185871627677343, −6.23506967327847336824045791812, −5.47776097722346093523183434876, −4.31751599962139242314372033028, −3.96878756842799062638779704978, −2.51983149173364353778170104074, −1.97811511861055873359095778766, −0.969908845788023342428147082539,
0.969908845788023342428147082539, 1.97811511861055873359095778766, 2.51983149173364353778170104074, 3.96878756842799062638779704978, 4.31751599962139242314372033028, 5.47776097722346093523183434876, 6.23506967327847336824045791812, 6.48570110285700185871627677343, 7.27071590728183928212578975815, 8.536396632589430445582287448371
