| L(s) = 1 | + 3.92·2-s + 11.4·4-s + 5·5-s − 6.21·7-s + 29.1·8-s + 19.6·10-s + 11·11-s − 25.1·13-s − 24.4·14-s + 68.6·16-s − 12.7·17-s + 57.0·20-s + 43.1·22-s + 25·25-s − 98.9·26-s − 70.9·28-s − 59.3·31-s + 153.·32-s − 50.0·34-s − 31.0·35-s + 145.·40-s + 50.7·43-s + 125.·44-s − 10.3·49-s + 98.1·50-s − 287.·52-s + 55·55-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.85·4-s + 5-s − 0.888·7-s + 3.63·8-s + 1.96·10-s + 11-s − 1.93·13-s − 1.74·14-s + 4.29·16-s − 0.750·17-s + 2.85·20-s + 1.96·22-s + 25-s − 3.80·26-s − 2.53·28-s − 1.91·31-s + 4.78·32-s − 1.47·34-s − 0.888·35-s + 3.63·40-s + 1.17·43-s + 2.85·44-s − 0.210·49-s + 1.96·50-s − 5.53·52-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.350570843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.350570843\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.92T + 4T^{2} \) |
| 7 | \( 1 + 6.21T + 49T^{2} \) |
| 13 | \( 1 + 25.1T + 169T^{2} \) |
| 17 | \( 1 + 12.7T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 59.3T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 59.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 107.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
| 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
−11.01859343838703469665940494053, −10.04918453089683149988692387192, −9.245816669305991156577307314818, −7.30019001640015893889543147935, −6.75358859304561070296495859205, −5.89193672390002890042401898799, −5.05551422380309320885100743536, −4.04103262350565372208806958951, −2.85046352767800731014950110064, −1.95913434305194413692663860787,
1.95913434305194413692663860787, 2.85046352767800731014950110064, 4.04103262350565372208806958951, 5.05551422380309320885100743536, 5.89193672390002890042401898799, 6.75358859304561070296495859205, 7.30019001640015893889543147935, 9.245816669305991156577307314818, 10.04918453089683149988692387192, 11.01859343838703469665940494053
