| L(s) = 1 | − 6.32·3-s − 18.9·7-s + 13.0·9-s − 12.6·11-s − 38·13-s − 34·17-s − 101.·19-s + 120.·21-s − 82.2·23-s + 88.5·27-s + 270·29-s − 341.·31-s + 80.0·33-s − 206·37-s + 240.·39-s − 270·41-s − 537.·43-s + 132.·47-s + 17·49-s + 215.·51-s + 258·53-s + 640.·57-s − 75.8·59-s − 250·61-s − 246.·63-s + 815.·67-s + 520·69-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.02·7-s + 0.481·9-s − 0.346·11-s − 0.810·13-s − 0.485·17-s − 1.22·19-s + 1.24·21-s − 0.745·23-s + 0.631·27-s + 1.72·29-s − 1.97·31-s + 0.422·33-s − 0.915·37-s + 0.986·39-s − 1.02·41-s − 1.90·43-s + 0.412·47-s + 0.0495·49-s + 0.590·51-s + 0.668·53-s + 1.48·57-s − 0.167·59-s − 0.524·61-s − 0.493·63-s + 1.48·67-s + 0.907·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2620683452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2620683452\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 6.32T + 27T^{2} \) |
| 7 | \( 1 + 18.9T + 343T^{2} \) |
| 11 | \( 1 + 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270T + 2.43e4T^{2} \) |
| 31 | \( 1 + 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206T + 5.06e4T^{2} \) |
| 41 | \( 1 + 270T + 6.89e4T^{2} \) |
| 43 | \( 1 + 537.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 132.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 258T + 1.48e5T^{2} \) |
| 59 | \( 1 + 75.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 815.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 890T + 7.04e5T^{2} \) |
| 97 | \( 1 - 254T + 9.12e5T^{2} \) |
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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
−10.23451933887939789669805235536, −9.131569080247708413189564627381, −8.179912953935986957600017457681, −6.84315727494272367078024195662, −6.49552164182341887712946692670, −5.46446838164856402271630170695, −4.69133480429537558933889286481, −3.43643670057893878187351628298, −2.11820551253215544693149834386, −0.28523851105051543404136060878,
0.28523851105051543404136060878, 2.11820551253215544693149834386, 3.43643670057893878187351628298, 4.69133480429537558933889286481, 5.46446838164856402271630170695, 6.49552164182341887712946692670, 6.84315727494272367078024195662, 8.179912953935986957600017457681, 9.131569080247708413189564627381, 10.23451933887939789669805235536
