| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2.73·7-s + 8-s + 9-s + 0.267·11-s + 12-s − 0.732·13-s + 2.73·14-s + 16-s − 4.19·17-s + 18-s − 19-s + 2.73·21-s + 0.267·22-s + 7.92·23-s + 24-s − 0.732·26-s + 27-s + 2.73·28-s + 1.73·29-s + 4.46·31-s + 32-s + 0.267·33-s − 4.19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.03·7-s + 0.353·8-s + 0.333·9-s + 0.0807·11-s + 0.288·12-s − 0.203·13-s + 0.730·14-s + 0.250·16-s − 1.01·17-s + 0.235·18-s − 0.229·19-s + 0.596·21-s + 0.0571·22-s + 1.65·23-s + 0.204·24-s − 0.143·26-s + 0.192·27-s + 0.516·28-s + 0.321·29-s + 0.801·31-s + 0.176·32-s + 0.0466·33-s − 0.719·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.188429733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.188429733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 23 | \( 1 - 7.92T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 0.267T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 9.12T + 97T^{2} \) |
| 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
−8.703008140941807890927914800494, −8.002699271760752799235842594880, −7.23740121736467875167359994927, −6.56468310042259988921502428767, −5.56196184613748288934216639159, −4.64079110519782434084833201718, −4.28386169049636262686734589583, −3.03625380417388351953711543354, −2.31845545892331003034549717024, −1.22425598400350502537696339361,
1.22425598400350502537696339361, 2.31845545892331003034549717024, 3.03625380417388351953711543354, 4.28386169049636262686734589583, 4.64079110519782434084833201718, 5.56196184613748288934216639159, 6.56468310042259988921502428767, 7.23740121736467875167359994927, 8.002699271760752799235842594880, 8.703008140941807890927914800494
