| L(s) = 1 | + 3-s − 7·7-s − 26·9-s − 37·11-s − 38·13-s + 35·17-s + 73·19-s − 7·21-s + 64·23-s − 53·27-s + 226·29-s + 108·31-s − 37·33-s + 360·37-s − 38·39-s + 279·41-s + 32·43-s − 222·47-s + 49·49-s + 35·51-s − 508·53-s + 73·57-s + 420·59-s − 610·61-s + 182·63-s + 825·67-s + 64·69-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.377·7-s − 0.962·9-s − 1.01·11-s − 0.810·13-s + 0.499·17-s + 0.881·19-s − 0.0727·21-s + 0.580·23-s − 0.377·27-s + 1.44·29-s + 0.625·31-s − 0.195·33-s + 1.59·37-s − 0.156·39-s + 1.06·41-s + 0.113·43-s − 0.688·47-s + 1/7·49-s + 0.0960·51-s − 1.31·53-s + 0.169·57-s + 0.926·59-s − 1.28·61-s + 0.363·63-s + 1.50·67-s + 0.111·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.566357504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566357504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 37 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 35 T + p^{3} T^{2} \) |
| 19 | \( 1 - 73 T + p^{3} T^{2} \) |
| 23 | \( 1 - 64 T + p^{3} T^{2} \) |
| 29 | \( 1 - 226 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 360 T + p^{3} T^{2} \) |
| 41 | \( 1 - 279 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 + 222 T + p^{3} T^{2} \) |
| 53 | \( 1 + 508 T + p^{3} T^{2} \) |
| 59 | \( 1 - 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 825 T + p^{3} T^{2} \) |
| 71 | \( 1 - 190 T + p^{3} T^{2} \) |
| 73 | \( 1 + 275 T + p^{3} T^{2} \) |
| 79 | \( 1 - 742 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1041 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1417 T + p^{3} T^{2} \) |
| 97 | \( 1 + 106 T + p^{3} T^{2} \) |
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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
−9.939390527362541444466363684451, −9.313806130303167106533252008479, −8.178086266475326942200228300178, −7.64136254148973610234278098907, −6.47020017262781506293635746092, −5.50843748637387560381254001801, −4.67877791751617010124032870426, −3.13559290612135832032999145355, −2.56295761583484233326493739199, −0.69279429554580685238235395427,
0.69279429554580685238235395427, 2.56295761583484233326493739199, 3.13559290612135832032999145355, 4.67877791751617010124032870426, 5.50843748637387560381254001801, 6.47020017262781506293635746092, 7.64136254148973610234278098907, 8.178086266475326942200228300178, 9.313806130303167106533252008479, 9.939390527362541444466363684451
