| L(s) = 1 | + 8.22·3-s − 17.2·5-s + 7·7-s + 40.6·9-s + 11·11-s + 62.9·13-s − 141.·15-s − 80.9·17-s − 44.9·19-s + 57.5·21-s − 55.0·23-s + 172.·25-s + 112.·27-s + 310.·29-s + 7.80·31-s + 90.4·33-s − 120.·35-s + 400.·37-s + 517.·39-s − 470.·41-s + 551.·43-s − 700.·45-s − 36.8·47-s + 49·49-s − 665.·51-s + 71.3·53-s − 189.·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s − 1.54·5-s + 0.377·7-s + 1.50·9-s + 0.301·11-s + 1.34·13-s − 2.44·15-s − 1.15·17-s − 0.542·19-s + 0.598·21-s − 0.499·23-s + 1.37·25-s + 0.799·27-s + 1.98·29-s + 0.0452·31-s + 0.477·33-s − 0.582·35-s + 1.78·37-s + 2.12·39-s − 1.79·41-s + 1.95·43-s − 2.32·45-s − 0.114·47-s + 0.142·49-s − 1.82·51-s + 0.184·53-s − 0.464·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.202807760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.202807760\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 8.22T + 27T^{2} \) |
| 5 | \( 1 + 17.2T + 125T^{2} \) |
| 13 | \( 1 - 62.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 310.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 7.80T + 2.97e4T^{2} \) |
| 37 | \( 1 - 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 551.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 36.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 71.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 623.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 326.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 617.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 719.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 244.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 927.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.849790351670044628987799076282, −8.510915582848661445476785074197, −8.020914551480091730848949218433, −7.15768589073573796689618258978, −6.27033104145863419430227816800, −4.48125745164405105358991625784, −4.07380158794476156860102961798, −3.23753431920576570393010535453, −2.22679013652312333182704119397, −0.863433353944770902059214139048,
0.863433353944770902059214139048, 2.22679013652312333182704119397, 3.23753431920576570393010535453, 4.07380158794476156860102961798, 4.48125745164405105358991625784, 6.27033104145863419430227816800, 7.15768589073573796689618258978, 8.020914551480091730848949218433, 8.510915582848661445476785074197, 8.849790351670044628987799076282
