| L(s) = 1 | − 3-s − 5-s − 2·9-s + 11-s − 13-s + 15-s − 17-s − 19-s + 4·23-s + 25-s + 5·27-s − 3·29-s − 11·31-s − 33-s − 4·37-s + 39-s − 8·41-s − 2·43-s + 2·45-s − 7·47-s − 7·49-s + 51-s − 9·53-s − 55-s + 57-s − 15·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 1.97·31-s − 0.174·33-s − 0.657·37-s + 0.160·39-s − 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.02·47-s − 49-s + 0.140·51-s − 1.23·53-s − 0.134·55-s + 0.132·57-s − 1.95·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.82931153941824, −14.41921986335555, −13.84939715582942, −13.22106372665696, −12.64980388107760, −12.35750469538528, −11.64150417826700, −11.26282116935338, −10.95762290601055, −10.37090487930812, −9.674715972657514, −9.095299230289838, −8.711136645909341, −8.086375741884646, −7.498157003638721, −6.912458138389881, −6.457222068524290, −5.843795956629130, −5.144682368045603, −4.888324176871634, −4.079785474565119, −3.325093075108273, −3.004861278952923, −1.928112034261597, −1.369228038540407, 0, 0,
1.369228038540407, 1.928112034261597, 3.004861278952923, 3.325093075108273, 4.079785474565119, 4.888324176871634, 5.144682368045603, 5.843795956629130, 6.457222068524290, 6.912458138389881, 7.498157003638721, 8.086375741884646, 8.711136645909341, 9.095299230289838, 9.674715972657514, 10.37090487930812, 10.95762290601055, 11.26282116935338, 11.64150417826700, 12.35750469538528, 12.64980388107760, 13.22106372665696, 13.84939715582942, 14.41921986335555, 14.82931153941824
