| L(s) = 1 | + 3-s + 2.65·5-s + 0.964·7-s + 9-s − 3.02·11-s + 5.02·13-s + 2.65·15-s + 3.53·17-s − 4.77·19-s + 0.964·21-s − 1.09·23-s + 2.03·25-s + 27-s + 1.56·29-s + 2.94·31-s − 3.02·33-s + 2.55·35-s + 1.47·37-s + 5.02·39-s − 10.6·41-s + 0.759·43-s + 2.65·45-s + 2.51·47-s − 6.07·49-s + 3.53·51-s + 0.376·53-s − 8.03·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.18·5-s + 0.364·7-s + 0.333·9-s − 0.913·11-s + 1.39·13-s + 0.684·15-s + 0.858·17-s − 1.09·19-s + 0.210·21-s − 0.227·23-s + 0.407·25-s + 0.192·27-s + 0.290·29-s + 0.528·31-s − 0.527·33-s + 0.432·35-s + 0.242·37-s + 0.805·39-s − 1.66·41-s + 0.115·43-s + 0.395·45-s + 0.367·47-s − 0.867·49-s + 0.495·51-s + 0.0517·53-s − 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396072310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396072310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 - 0.964T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 0.759T + 43T^{2} \) |
| 47 | \( 1 - 2.51T + 47T^{2} \) |
| 53 | \( 1 - 0.376T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 + 1.76T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
| 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
−10.26801696154119804171695244731, −9.445820501506822595478541452968, −8.463521555108328253766337996350, −7.997900560160534049477424680541, −6.66545025064977816140724901402, −5.88518483527926250900517213249, −4.98249291290856679379388056001, −3.70209917607951266001038280678, −2.52147684464407657087148595927, −1.49024632447679905921558947112,
1.49024632447679905921558947112, 2.52147684464407657087148595927, 3.70209917607951266001038280678, 4.98249291290856679379388056001, 5.88518483527926250900517213249, 6.66545025064977816140724901402, 7.997900560160534049477424680541, 8.463521555108328253766337996350, 9.445820501506822595478541452968, 10.26801696154119804171695244731
