| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 3·13-s + 15-s + 17-s + 5·19-s + 21-s + 7·23-s − 4·25-s − 27-s − 6·29-s + 4·31-s + 33-s + 35-s − 2·37-s + 3·39-s − 3·41-s − 7·43-s − 45-s − 4·47-s + 49-s − 51-s + 55-s − 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.258·15-s + 0.242·17-s + 1.14·19-s + 0.218·21-s + 1.45·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.169·35-s − 0.328·37-s + 0.480·39-s − 0.468·41-s − 1.06·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.134·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9758347442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9758347442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.88396101199303644248386128762, −7.41280636873560990177498661283, −6.78971231160232287423313085988, −5.93457588676100219163990903083, −5.16238410807211888869004576932, −4.69206808958976880214106295888, −3.56988859770934196455165706202, −2.98410148036278223731071512199, −1.75992872328618883663801041609, −0.53596060817849207730111837092,
0.53596060817849207730111837092, 1.75992872328618883663801041609, 2.98410148036278223731071512199, 3.56988859770934196455165706202, 4.69206808958976880214106295888, 5.16238410807211888869004576932, 5.93457588676100219163990903083, 6.78971231160232287423313085988, 7.41280636873560990177498661283, 7.88396101199303644248386128762
