| L(s) = 1 | + 2-s + 2.19·3-s + 4-s + 2.75·5-s + 2.19·6-s + 8-s + 1.81·9-s + 2.75·10-s + 0.941·11-s + 2.19·12-s + 4.37·13-s + 6.04·15-s + 16-s + 1.93·17-s + 1.81·18-s − 2.67·19-s + 2.75·20-s + 0.941·22-s − 4.18·23-s + 2.19·24-s + 2.58·25-s + 4.37·26-s − 2.59·27-s − 0.145·29-s + 6.04·30-s + 7.42·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 1.23·5-s + 0.896·6-s + 0.353·8-s + 0.605·9-s + 0.870·10-s + 0.284·11-s + 0.633·12-s + 1.21·13-s + 1.56·15-s + 0.250·16-s + 0.469·17-s + 0.428·18-s − 0.614·19-s + 0.615·20-s + 0.200·22-s − 0.873·23-s + 0.448·24-s + 0.517·25-s + 0.858·26-s − 0.499·27-s − 0.0270·29-s + 1.10·30-s + 1.33·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.324047575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.324047575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 11 | \( 1 - 0.941T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 - 7.42T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 5.20T + 59T^{2} \) |
| 61 | \( 1 + 1.15T + 61T^{2} \) |
| 67 | \( 1 - 0.684T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
| 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
−8.544598535207882885543611195451, −7.81963653476777939157925689308, −6.84606771914093120289689608139, −6.07991166388938920668005465021, −5.65686541829663677946372982050, −4.50587376359069029493106518782, −3.70092088724940738650768210562, −2.99420540642999989891363029577, −2.12425722154318921575100508568, −1.45408118511750262018549759718,
1.45408118511750262018549759718, 2.12425722154318921575100508568, 2.99420540642999989891363029577, 3.70092088724940738650768210562, 4.50587376359069029493106518782, 5.65686541829663677946372982050, 6.07991166388938920668005465021, 6.84606771914093120289689608139, 7.81963653476777939157925689308, 8.544598535207882885543611195451
