| L(s) = 1 | − 3-s − 7-s + 9-s + 13-s + 5·19-s + 21-s + 6·23-s − 27-s + 4·29-s − 5·31-s + 3·37-s − 39-s + 6·41-s − 7·43-s − 4·47-s + 49-s + 10·53-s − 5·57-s − 10·59-s − 5·61-s − 63-s + 11·67-s − 6·69-s − 6·71-s − 11·73-s − 7·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.277·13-s + 1.14·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.742·29-s − 0.898·31-s + 0.493·37-s − 0.160·39-s + 0.937·41-s − 1.06·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.662·57-s − 1.30·59-s − 0.640·61-s − 0.125·63-s + 1.34·67-s − 0.722·69-s − 0.712·71-s − 1.28·73-s − 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.952256597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.952256597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.83034453217866, −12.37799109465960, −11.79638968910044, −11.55670607985269, −10.99257470374988, −10.57265059932810, −10.12071760492573, −9.605060300648248, −9.116617268084034, −8.816785763103229, −8.053799036594459, −7.614538988531866, −7.097448722412713, −6.704153247298180, −6.176910124157231, −5.606029399276724, −5.271664248030578, −4.652090353750039, −4.205561441162647, −3.423695141332760, −3.102600252991985, −2.466940967535418, −1.605302707165193, −1.081851695999273, −0.4469493698976472,
0.4469493698976472, 1.081851695999273, 1.605302707165193, 2.466940967535418, 3.102600252991985, 3.423695141332760, 4.205561441162647, 4.652090353750039, 5.271664248030578, 5.606029399276724, 6.176910124157231, 6.704153247298180, 7.097448722412713, 7.614538988531866, 8.053799036594459, 8.816785763103229, 9.116617268084034, 9.605060300648248, 10.12071760492573, 10.57265059932810, 10.99257470374988, 11.55670607985269, 11.79638968910044, 12.37799109465960, 12.83034453217866
