L(s) = 1 | − 4.46·2-s + 11.9·4-s + 5·5-s − 17.6·8-s − 22.3·10-s + 56.5·11-s + 40.9·13-s − 16.6·16-s − 2.18·17-s + 16.4·19-s + 59.7·20-s − 252.·22-s + 155.·23-s + 25·25-s − 183.·26-s + 6.26·29-s + 168.·31-s + 215.·32-s + 9.77·34-s − 37.1·37-s − 73.5·38-s − 88.4·40-s + 266.·41-s − 14.6·43-s + 676.·44-s − 693.·46-s + 169.·47-s + ⋯ |
L(s) = 1 | − 1.57·2-s + 1.49·4-s + 0.447·5-s − 0.781·8-s − 0.706·10-s + 1.54·11-s + 0.873·13-s − 0.260·16-s − 0.0312·17-s + 0.198·19-s + 0.668·20-s − 2.44·22-s + 1.40·23-s + 0.200·25-s − 1.38·26-s + 0.0400·29-s + 0.977·31-s + 1.19·32-s + 0.0493·34-s − 0.165·37-s − 0.314·38-s − 0.349·40-s + 1.01·41-s − 0.0519·43-s + 2.31·44-s − 2.22·46-s + 0.527·47-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
\(L(2)\) |
\(\approx\) |
\(1.547054906\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547054906\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.46T + 8T^{2} \) |
| 11 | \( 1 - 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.26T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 934.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 752.T + 9.12e5T^{2} \) |
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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
−8.904191125128153463859507862445, −8.199139167560110172214190962401, −7.20955453496508137556154116033, −6.63096786233612343384621525467, −5.93944518125269455714336230501, −4.68175142394405236460018311006, −3.62148541064919988017290443933, −2.44811561400492220998719715751, −1.32733214524818782582278840857, −0.846889252929135547993845561864,
0.846889252929135547993845561864, 1.32733214524818782582278840857, 2.44811561400492220998719715751, 3.62148541064919988017290443933, 4.68175142394405236460018311006, 5.93944518125269455714336230501, 6.63096786233612343384621525467, 7.20955453496508137556154116033, 8.199139167560110172214190962401, 8.904191125128153463859507862445