| L(s) = 1 | − 3.68·2-s + 5.59·4-s − 11.8·5-s + 8.85·8-s + 43.7·10-s − 22.2·11-s − 75.1·13-s − 77.4·16-s + 74.1·17-s + 7.41·19-s − 66.4·20-s + 81.9·22-s + 205.·23-s + 15.9·25-s + 276.·26-s − 148.·29-s + 164.·31-s + 214.·32-s − 273.·34-s − 205.·37-s − 27.3·38-s − 105.·40-s + 83.9·41-s − 368.·43-s − 124.·44-s − 758.·46-s + 98.3·47-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.699·4-s − 1.06·5-s + 0.391·8-s + 1.38·10-s − 0.608·11-s − 1.60·13-s − 1.21·16-s + 1.05·17-s + 0.0895·19-s − 0.743·20-s + 0.793·22-s + 1.86·23-s + 0.127·25-s + 2.08·26-s − 0.949·29-s + 0.952·31-s + 1.18·32-s − 1.37·34-s − 0.911·37-s − 0.116·38-s − 0.415·40-s + 0.319·41-s − 1.30·43-s − 0.426·44-s − 2.43·46-s + 0.305·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.68T + 8T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 11 | \( 1 + 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.41T + 6.85e3T^{2} \) |
| 23 | \( 1 - 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 83.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 98.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 293.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 509.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 121.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 682.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 669.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−8.716825033313170160833453516212, −8.147539251295607173850099776354, −7.22934923458286036095388404925, −7.15856371889363806204943324487, −5.36298718312043907799187142132, −4.66973403249598545827425700627, −3.44245629685745685875045094732, −2.33702165415097516486525467925, −0.910064134066559161033882164644, 0,
0.910064134066559161033882164644, 2.33702165415097516486525467925, 3.44245629685745685875045094732, 4.66973403249598545827425700627, 5.36298718312043907799187142132, 7.15856371889363806204943324487, 7.22934923458286036095388404925, 8.147539251295607173850099776354, 8.716825033313170160833453516212
