| L(s) = 1 | + 2.97·2-s − 7.85·3-s − 23.1·4-s + 25·5-s − 23.3·6-s + 148.·7-s − 163.·8-s − 181.·9-s + 74.2·10-s + 121·11-s + 181.·12-s − 569.·13-s + 441.·14-s − 196.·15-s + 254.·16-s − 1.32e3·17-s − 538.·18-s + 361·19-s − 579.·20-s − 1.16e3·21-s + 359.·22-s − 3.50e3·23-s + 1.28e3·24-s + 625·25-s − 1.69e3·26-s + 3.33e3·27-s − 3.44e3·28-s + ⋯ |
| L(s) = 1 | + 0.525·2-s − 0.503·3-s − 0.724·4-s + 0.447·5-s − 0.264·6-s + 1.14·7-s − 0.905·8-s − 0.746·9-s + 0.234·10-s + 0.301·11-s + 0.364·12-s − 0.935·13-s + 0.601·14-s − 0.225·15-s + 0.248·16-s − 1.11·17-s − 0.391·18-s + 0.229·19-s − 0.323·20-s − 0.576·21-s + 0.158·22-s − 1.38·23-s + 0.455·24-s + 0.200·25-s − 0.491·26-s + 0.879·27-s − 0.829·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.369890760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369890760\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 2.97T + 32T^{2} \) |
| 3 | \( 1 + 7.85T + 243T^{2} \) |
| 7 | \( 1 - 148.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 569.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.08e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 692.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.91e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.75e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.19e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.69e5T + 8.58e9T^{2} \) |
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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
−9.150747766322410939127412470782, −8.442280932543317757848665128599, −7.58183999805840318821642528358, −6.28143642506924559701051126017, −5.68408703408116808637576883782, −4.78242743203824644462326577594, −4.33181170730375194230990223790, −2.91041674040918508967522504038, −1.84883252821370402824943444400, −0.47134389747050991972225416433,
0.47134389747050991972225416433, 1.84883252821370402824943444400, 2.91041674040918508967522504038, 4.33181170730375194230990223790, 4.78242743203824644462326577594, 5.68408703408116808637576883782, 6.28143642506924559701051126017, 7.58183999805840318821642528358, 8.442280932543317757848665128599, 9.150747766322410939127412470782
