| L(s) = 1 | − 3-s + 9-s − 11-s − 5·13-s + 6·17-s + 19-s + 2·23-s − 27-s − 6·29-s + 4·31-s + 33-s − 37-s + 5·39-s − 6·41-s + 4·43-s − 6·47-s − 6·51-s − 57-s − 61-s − 5·67-s − 2·69-s − 6·71-s + 9·73-s − 5·79-s + 81-s + 6·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 1.45·17-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.164·37-s + 0.800·39-s − 0.937·41-s + 0.609·43-s − 0.875·47-s − 0.840·51-s − 0.132·57-s − 0.128·61-s − 0.610·67-s − 0.240·69-s − 0.712·71-s + 1.05·73-s − 0.562·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| 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
−12.68147200325502, −12.32743726716186, −12.01194056842978, −11.53424255251044, −11.08145159491306, −10.51595813056604, −10.07958774049998, −9.771547745589162, −9.344051317120545, −8.764775557016374, −8.039734166129801, −7.769225695108205, −7.249304738711245, −6.910612196537135, −6.243563567390052, −5.745367595874482, −5.205227429698029, −5.012703250598789, −4.380054786286855, −3.746019559546041, −3.144989953262496, −2.717626655211220, −1.967639787749825, −1.403109335539524, −0.6718720970797470, 0,
0.6718720970797470, 1.403109335539524, 1.967639787749825, 2.717626655211220, 3.144989953262496, 3.746019559546041, 4.380054786286855, 5.012703250598789, 5.205227429698029, 5.745367595874482, 6.243563567390052, 6.910612196537135, 7.249304738711245, 7.769225695108205, 8.039734166129801, 8.764775557016374, 9.344051317120545, 9.771547745589162, 10.07958774049998, 10.51595813056604, 11.08145159491306, 11.53424255251044, 12.01194056842978, 12.32743726716186, 12.68147200325502
