| L(s) = 1 | − 5.20·2-s + 19.0·4-s − 5·5-s − 57.6·8-s + 26.0·10-s − 56.6·11-s + 43.4·13-s + 147.·16-s − 39.8·17-s − 52.3·19-s − 95.3·20-s + 294.·22-s + 53.5·23-s + 25·25-s − 225.·26-s − 49.6·29-s + 73.7·31-s − 305.·32-s + 207.·34-s − 307.·37-s + 272.·38-s + 288.·40-s + 292.·41-s − 365.·43-s − 1.08e3·44-s − 278.·46-s − 442.·47-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.38·4-s − 0.447·5-s − 2.54·8-s + 0.822·10-s − 1.55·11-s + 0.926·13-s + 2.30·16-s − 0.568·17-s − 0.632·19-s − 1.06·20-s + 2.85·22-s + 0.485·23-s + 0.200·25-s − 1.70·26-s − 0.317·29-s + 0.427·31-s − 1.68·32-s + 1.04·34-s − 1.36·37-s + 1.16·38-s + 1.13·40-s + 1.11·41-s − 1.29·43-s − 3.70·44-s − 0.892·46-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3184890611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3184890611\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.20T + 8T^{2} \) |
| 11 | \( 1 + 56.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 25.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 511.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 134.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 409.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 926.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 296.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 488.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 475.T + 9.12e5T^{2} \) |
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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
−8.444648816804266169228558962011, −8.295984246461963348823047958578, −7.39586293797013354895189064437, −6.74260874749952116528547544882, −5.88393035501710146999830574013, −4.78121388250290639271504972806, −3.39608942089151859182867711145, −2.49486395269559300339835338332, −1.53246253478909874278807853100, −0.33759025064208613497695651845,
0.33759025064208613497695651845, 1.53246253478909874278807853100, 2.49486395269559300339835338332, 3.39608942089151859182867711145, 4.78121388250290639271504972806, 5.88393035501710146999830574013, 6.74260874749952116528547544882, 7.39586293797013354895189064437, 8.295984246461963348823047958578, 8.444648816804266169228558962011
