| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 0.723·11-s + 12-s + 4.80·13-s + 14-s + 15-s + 16-s − 5.52·17-s + 18-s − 3.52·19-s + 20-s + 21-s + 0.723·22-s + 23-s + 24-s + 25-s + 4.80·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.218·11-s + 0.288·12-s + 1.33·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.34·17-s + 0.235·18-s − 0.808·19-s + 0.223·20-s + 0.218·21-s + 0.154·22-s + 0.208·23-s + 0.204·24-s + 0.200·25-s + 0.941·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.828745942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.828745942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 0.723T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 9.47T + 41T^{2} \) |
| 43 | \( 1 - 3.66T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 + 7.98T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + 4.97T + 83T^{2} \) |
| 89 | \( 1 - 0.416T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
| 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
−8.267142404669091367086254753567, −7.55881620142382041717020384546, −6.58420357567801884635829582730, −6.19971135465577484828825622613, −5.33678665621750305567197372049, −4.26281329702168028288025760893, −4.02630183575220137649727908530, −2.80256968467775553129932742462, −2.15205047479096589832105140153, −1.14545532036332069967585404631,
1.14545532036332069967585404631, 2.15205047479096589832105140153, 2.80256968467775553129932742462, 4.02630183575220137649727908530, 4.26281329702168028288025760893, 5.33678665621750305567197372049, 6.19971135465577484828825622613, 6.58420357567801884635829582730, 7.55881620142382041717020384546, 8.267142404669091367086254753567
