| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s + 4·11-s − 54·13-s + 14·17-s + 92·19-s − 21·21-s + 152·23-s − 27·27-s − 106·29-s − 144·31-s − 12·33-s − 158·37-s + 162·39-s − 390·41-s + 508·43-s + 528·47-s + 49·49-s − 42·51-s − 606·53-s − 276·57-s − 364·59-s + 678·61-s + 63·63-s − 844·67-s − 456·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.109·11-s − 1.15·13-s + 0.199·17-s + 1.11·19-s − 0.218·21-s + 1.37·23-s − 0.192·27-s − 0.678·29-s − 0.834·31-s − 0.0633·33-s − 0.702·37-s + 0.665·39-s − 1.48·41-s + 1.80·43-s + 1.63·47-s + 1/7·49-s − 0.115·51-s − 1.57·53-s − 0.641·57-s − 0.803·59-s + 1.42·61-s + 0.125·63-s − 1.53·67-s − 0.795·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.647786247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.647786247\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 508 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 606 T + p^{3} T^{2} \) |
| 59 | \( 1 + 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 678 T + p^{3} T^{2} \) |
| 67 | \( 1 + 844 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−8.973717165393027095650813641124, −7.61829011758028150733323460229, −7.37926271137014986671725390496, −6.41013498270428555975300903561, −5.29612359907441728252454139324, −5.04884284403760732411064517662, −3.89165353806744324855294226971, −2.85917309412337005446459649264, −1.69854703260050036206031523288, −0.61429701986968373460264030841,
0.61429701986968373460264030841, 1.69854703260050036206031523288, 2.85917309412337005446459649264, 3.89165353806744324855294226971, 5.04884284403760732411064517662, 5.29612359907441728252454139324, 6.41013498270428555975300903561, 7.37926271137014986671725390496, 7.61829011758028150733323460229, 8.973717165393027095650813641124
