| L(s) = 1 | − 0.845·2-s + 3·3-s − 7.28·4-s + 16.8·5-s − 2.53·6-s − 7·7-s + 12.9·8-s + 9·9-s − 14.2·10-s + 34.9·11-s − 21.8·12-s + 65.0·13-s + 5.91·14-s + 50.4·15-s + 47.3·16-s − 125.·17-s − 7.60·18-s − 47.1·19-s − 122.·20-s − 21·21-s − 29.5·22-s + 23·23-s + 38.7·24-s + 157.·25-s − 55.0·26-s + 27·27-s + 50.9·28-s + ⋯ |
| L(s) = 1 | − 0.298·2-s + 0.577·3-s − 0.910·4-s + 1.50·5-s − 0.172·6-s − 0.377·7-s + 0.571·8-s + 0.333·9-s − 0.449·10-s + 0.959·11-s − 0.525·12-s + 1.38·13-s + 0.112·14-s + 0.868·15-s + 0.739·16-s − 1.78·17-s − 0.0996·18-s − 0.569·19-s − 1.36·20-s − 0.218·21-s − 0.286·22-s + 0.208·23-s + 0.329·24-s + 1.26·25-s − 0.415·26-s + 0.192·27-s + 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.338623770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.338623770\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 0.845T + 8T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 11 | \( 1 - 34.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 210.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 56.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 412.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 534.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 213.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 59.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 28.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 96.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 134.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 905.T + 9.12e5T^{2} \) |
| 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
−10.30960368737076960170433735432, −9.391403865716038098171310969249, −9.001037078156228276455893436047, −8.298985196694790072424288790690, −6.65149130667629782961669241271, −6.11752539594507712107011795695, −4.72605760045700993763245037526, −3.73949597102992914178232793014, −2.23426488588757368576901254941, −1.06066840347861899393324104687,
1.06066840347861899393324104687, 2.23426488588757368576901254941, 3.73949597102992914178232793014, 4.72605760045700993763245037526, 6.11752539594507712107011795695, 6.65149130667629782961669241271, 8.298985196694790072424288790690, 9.001037078156228276455893436047, 9.391403865716038098171310969249, 10.30960368737076960170433735432
