| L(s) = 1 | − 2·3-s + 7-s + 9-s + 2·11-s + 4·13-s + 2·17-s + 4·19-s − 2·21-s + 23-s + 4·27-s − 2·29-s + 4·31-s − 4·33-s + 8·37-s − 8·39-s − 2·41-s + 10·43-s − 12·47-s + 49-s − 4·51-s + 4·53-s − 8·57-s − 10·59-s − 14·61-s + 63-s + 10·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 0.436·21-s + 0.208·23-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 1.31·37-s − 1.28·39-s − 0.312·41-s + 1.52·43-s − 1.75·47-s + 1/7·49-s − 0.560·51-s + 0.549·53-s − 1.05·57-s − 1.30·59-s − 1.79·61-s + 0.125·63-s + 1.22·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.191207977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191207977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.19967849388189, −13.74884701113445, −13.22687408508393, −12.51993070140803, −12.18090610767332, −11.56063388116840, −11.29388759243392, −10.84816643593649, −10.36481620606946, −9.538277850379679, −9.320885642504927, −8.522471106103635, −7.983372745539259, −7.525383609723827, −6.693048672544801, −6.322709451472081, −5.830565162702454, −5.326056077191358, −4.723024747595482, −4.176935225462696, −3.424736145325238, −2.877247622714326, −1.824294941698809, −1.100870512764307, −0.6512387503193555,
0.6512387503193555, 1.100870512764307, 1.824294941698809, 2.877247622714326, 3.424736145325238, 4.176935225462696, 4.723024747595482, 5.326056077191358, 5.830565162702454, 6.322709451472081, 6.693048672544801, 7.525383609723827, 7.983372745539259, 8.522471106103635, 9.320885642504927, 9.538277850379679, 10.36481620606946, 10.84816643593649, 11.29388759243392, 11.56063388116840, 12.18090610767332, 12.51993070140803, 13.22687408508393, 13.74884701113445, 14.19967849388189
