| L(s) = 1 | + 2.53·2-s + 4.41·4-s + 3.22·7-s + 6.10·8-s + 3.10·11-s + 2.18·13-s + 8.17·14-s + 6.63·16-s + 3·17-s + 0.0418·19-s + 7.86·22-s + 6.10·23-s + 5.53·26-s + 14.2·28-s − 6.57·29-s − 6.22·31-s + 4.59·32-s + 7.59·34-s − 3.59·37-s + 0.106·38-s − 7.70·41-s + 0.588·43-s + 13.7·44-s + 15.4·46-s − 9.66·47-s + 3.41·49-s + 9.63·52-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s + 1.21·7-s + 2.15·8-s + 0.936·11-s + 0.605·13-s + 2.18·14-s + 1.65·16-s + 0.727·17-s + 0.00961·19-s + 1.67·22-s + 1.27·23-s + 1.08·26-s + 2.69·28-s − 1.22·29-s − 1.11·31-s + 0.812·32-s + 1.30·34-s − 0.591·37-s + 0.0172·38-s − 1.20·41-s + 0.0897·43-s + 2.06·44-s + 2.27·46-s − 1.40·47-s + 0.487·49-s + 1.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.322118212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.322118212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 0.0418T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 + 3.59T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 0.588T + 43T^{2} \) |
| 47 | \( 1 + 9.66T + 47T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.23T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.50T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.79502422615483839847964880577, −7.08927211780181096023532058359, −6.53152243492050434574558962285, −5.57146046638486826574026315769, −5.23659989248036097669727119642, −4.46693832214843610909082651515, −3.70410729153630710114023959693, −3.20319400381982646828516155318, −1.94507418025516862670667619824, −1.37293628271581631243672623373,
1.37293628271581631243672623373, 1.94507418025516862670667619824, 3.20319400381982646828516155318, 3.70410729153630710114023959693, 4.46693832214843610909082651515, 5.23659989248036097669727119642, 5.57146046638486826574026315769, 6.53152243492050434574558962285, 7.08927211780181096023532058359, 7.79502422615483839847964880577
