| L(s) = 1 | − 2-s + 0.0287·3-s + 4-s + 1.06·5-s − 0.0287·6-s − 7-s − 8-s − 2.99·9-s − 1.06·10-s + 2.45·11-s + 0.0287·12-s − 6.62·13-s + 14-s + 0.0305·15-s + 16-s + 4.12·17-s + 2.99·18-s − 6.10·19-s + 1.06·20-s − 0.0287·21-s − 2.45·22-s − 5.32·23-s − 0.0287·24-s − 3.87·25-s + 6.62·26-s − 0.172·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0165·3-s + 0.5·4-s + 0.475·5-s − 0.0117·6-s − 0.377·7-s − 0.353·8-s − 0.999·9-s − 0.335·10-s + 0.741·11-s + 0.00829·12-s − 1.83·13-s + 0.267·14-s + 0.00787·15-s + 0.250·16-s + 1.00·17-s + 0.706·18-s − 1.40·19-s + 0.237·20-s − 0.00626·21-s − 0.524·22-s − 1.11·23-s − 0.00586·24-s − 0.774·25-s + 1.29·26-s − 0.0331·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8468307598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8468307598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0287T + 3T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 - 9.64T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 + 8.80T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.145741922826750712103262372052, −7.49306738301382425293077249839, −6.63508493350725065772252193910, −6.07529378732125617602890863375, −5.43244627302097964339300154283, −4.43925962443540594902459621831, −3.47459871674822685228299103197, −2.49086899465120371063305842673, −1.98346733470570395259690044615, −0.50653418709760305676460211404,
0.50653418709760305676460211404, 1.98346733470570395259690044615, 2.49086899465120371063305842673, 3.47459871674822685228299103197, 4.43925962443540594902459621831, 5.43244627302097964339300154283, 6.07529378732125617602890863375, 6.63508493350725065772252193910, 7.49306738301382425293077249839, 8.145741922826750712103262372052
