| L(s) = 1 | − 5.24·2-s + 19.5·4-s + 10.0·5-s + 0.804·7-s − 60.5·8-s − 52.9·10-s + 0.829·11-s − 86.7·13-s − 4.22·14-s + 161.·16-s + 50.2·17-s + 121.·19-s + 197.·20-s − 4.35·22-s − 76.0·23-s − 23.0·25-s + 455.·26-s + 15.7·28-s + 40.2·29-s − 81.0·31-s − 361.·32-s − 263.·34-s + 8.12·35-s − 321.·37-s − 635.·38-s − 610.·40-s − 147.·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.44·4-s + 0.902·5-s + 0.0434·7-s − 2.67·8-s − 1.67·10-s + 0.0227·11-s − 1.85·13-s − 0.0805·14-s + 2.51·16-s + 0.716·17-s + 1.46·19-s + 2.20·20-s − 0.0421·22-s − 0.689·23-s − 0.184·25-s + 3.43·26-s + 0.106·28-s + 0.258·29-s − 0.469·31-s − 1.99·32-s − 1.32·34-s + 0.0392·35-s − 1.42·37-s − 2.71·38-s − 2.41·40-s − 0.562·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8286912066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8286912066\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 5.24T + 8T^{2} \) |
| 5 | \( 1 - 10.0T + 125T^{2} \) |
| 7 | \( 1 - 0.804T + 343T^{2} \) |
| 11 | \( 1 - 0.829T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 321.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 110.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 817.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 492.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 395.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 210.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 879.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 461.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 372.T + 9.12e5T^{2} \) |
| 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
−8.919689175402494784115086046389, −7.948124955636567166379509212751, −7.41150596797329233943019332816, −6.77426492917819487099938042667, −5.75624540643917010983929620323, −5.07112183971776368690799240124, −3.31399827955287438590322494911, −2.33416974709948998510548858105, −1.66809544324431033929281531996, −0.54208577956489765223165170249,
0.54208577956489765223165170249, 1.66809544324431033929281531996, 2.33416974709948998510548858105, 3.31399827955287438590322494911, 5.07112183971776368690799240124, 5.75624540643917010983929620323, 6.77426492917819487099938042667, 7.41150596797329233943019332816, 7.948124955636567166379509212751, 8.919689175402494784115086046389
