| L(s) = 1 | − 0.881·3-s + 5-s + 3.19·7-s − 2.22·9-s − 2.49·11-s + 4.91·13-s − 0.881·15-s − 1.58·17-s − 3.07·19-s − 2.81·21-s + 6.96·23-s + 25-s + 4.60·27-s + 7.93·29-s + 7.30·31-s + 2.19·33-s + 3.19·35-s − 10.3·37-s − 4.33·39-s − 8.96·41-s − 8.99·43-s − 2.22·45-s + 0.801·47-s + 3.19·49-s + 1.39·51-s + 6.89·53-s − 2.49·55-s + ⋯ |
| L(s) = 1 | − 0.508·3-s + 0.447·5-s + 1.20·7-s − 0.741·9-s − 0.751·11-s + 1.36·13-s − 0.227·15-s − 0.383·17-s − 0.705·19-s − 0.614·21-s + 1.45·23-s + 0.200·25-s + 0.885·27-s + 1.47·29-s + 1.31·31-s + 0.382·33-s + 0.539·35-s − 1.70·37-s − 0.693·39-s − 1.40·41-s − 1.37·43-s − 0.331·45-s + 0.116·47-s + 0.457·49-s + 0.195·51-s + 0.946·53-s − 0.336·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068220277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068220277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.881T + 3T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 - 0.801T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 - 3.86T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.08T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 0.834T + 73T^{2} \) |
| 79 | \( 1 + 7.84T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
| 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
−8.139580929170725192688692870390, −6.81042493109514644294224496393, −6.59193967707120002992463990957, −5.57844347488841632957917052754, −5.11585067099842334894861360204, −4.58769449534674387213245929579, −3.44086167475653534182606355991, −2.62171381313782916690171736190, −1.68663056552152898811152111872, −0.75742921235528968662586393691,
0.75742921235528968662586393691, 1.68663056552152898811152111872, 2.62171381313782916690171736190, 3.44086167475653534182606355991, 4.58769449534674387213245929579, 5.11585067099842334894861360204, 5.57844347488841632957917052754, 6.59193967707120002992463990957, 6.81042493109514644294224496393, 8.139580929170725192688692870390
