| L(s) = 1 | − 2·3-s − 6·7-s − 23·9-s − 60·11-s + 50·13-s + 30·17-s − 40·19-s + 12·21-s − 178·23-s + 100·27-s − 166·29-s + 20·31-s + 120·33-s + 10·37-s − 100·39-s − 250·41-s + 142·43-s − 214·47-s − 307·49-s − 60·51-s + 490·53-s + 80·57-s + 800·59-s − 250·61-s + 138·63-s − 774·67-s + 356·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.323·7-s − 0.851·9-s − 1.64·11-s + 1.06·13-s + 0.428·17-s − 0.482·19-s + 0.124·21-s − 1.61·23-s + 0.712·27-s − 1.06·29-s + 0.115·31-s + 0.633·33-s + 0.0444·37-s − 0.410·39-s − 0.952·41-s + 0.503·43-s − 0.664·47-s − 0.895·49-s − 0.164·51-s + 1.26·53-s + 0.185·57-s + 1.76·59-s − 0.524·61-s + 0.275·63-s − 1.41·67-s + 0.621·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7531880770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7531880770\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 178 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 10 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 - 142 T + p^{3} T^{2} \) |
| 47 | \( 1 + 214 T + p^{3} T^{2} \) |
| 53 | \( 1 - 490 T + p^{3} T^{2} \) |
| 59 | \( 1 - 800 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 774 T + p^{3} T^{2} \) |
| 71 | \( 1 - 100 T + p^{3} T^{2} \) |
| 73 | \( 1 - 230 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 982 T + p^{3} T^{2} \) |
| 89 | \( 1 - 874 T + p^{3} T^{2} \) |
| 97 | \( 1 - 310 T + p^{3} T^{2} \) |
| 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
−8.907702837367720328432740518269, −8.226588177259712087192761513742, −7.60048291743330107364646457976, −6.39744904284360417470975965381, −5.78121775745978168209857278042, −5.14473643091440164437368972636, −3.91437037206967384233151091114, −3.00285496028347574989185001842, −1.97553125537073723217644701003, −0.40476232291444654274709336159,
0.40476232291444654274709336159, 1.97553125537073723217644701003, 3.00285496028347574989185001842, 3.91437037206967384233151091114, 5.14473643091440164437368972636, 5.78121775745978168209857278042, 6.39744904284360417470975965381, 7.60048291743330107364646457976, 8.226588177259712087192761513742, 8.907702837367720328432740518269
