| L(s) = 1 | + 4.39·2-s + 7.85·3-s + 11.3·4-s + 5.85·5-s + 34.5·6-s + 12.3·7-s + 14.6·8-s + 34.6·9-s + 25.7·10-s + 50.9·11-s + 88.9·12-s − 72.3·13-s + 54.4·14-s + 45.9·15-s − 26.3·16-s + 72.1·17-s + 152.·18-s + 47.7·19-s + 66.2·20-s + 97.3·21-s + 223.·22-s + 194.·23-s + 114.·24-s − 90.7·25-s − 318.·26-s + 60.4·27-s + 140.·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.51·3-s + 1.41·4-s + 0.523·5-s + 2.34·6-s + 0.669·7-s + 0.645·8-s + 1.28·9-s + 0.813·10-s + 1.39·11-s + 2.13·12-s − 1.54·13-s + 1.04·14-s + 0.791·15-s − 0.412·16-s + 1.02·17-s + 1.99·18-s + 0.576·19-s + 0.740·20-s + 1.01·21-s + 2.16·22-s + 1.76·23-s + 0.975·24-s − 0.726·25-s − 2.39·26-s + 0.430·27-s + 0.947·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.25565020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.25565020\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 4.39T + 8T^{2} \) |
| 3 | \( 1 - 7.85T + 27T^{2} \) |
| 5 | \( 1 - 5.85T + 125T^{2} \) |
| 7 | \( 1 - 12.3T + 343T^{2} \) |
| 11 | \( 1 - 50.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 485.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 20.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 65.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 330.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 522.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 155.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 543.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 781.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 515.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 901.T + 9.12e5T^{2} \) |
| 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
−9.059036239601626526266278170122, −7.931422488709464024089813900168, −7.22276112409326050918639228476, −6.48567024811643633543270846946, −5.27427848976790285543129200219, −4.80255440785328257936059516161, −3.73077052848216322826384001462, −3.14678647874950527038585172134, −2.27991804291914812483669213366, −1.42080795960944746245752474043,
1.42080795960944746245752474043, 2.27991804291914812483669213366, 3.14678647874950527038585172134, 3.73077052848216322826384001462, 4.80255440785328257936059516161, 5.27427848976790285543129200219, 6.48567024811643633543270846946, 7.22276112409326050918639228476, 7.931422488709464024089813900168, 9.059036239601626526266278170122
