| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 13-s + 2·14-s + 16-s − 2·17-s − 2·19-s + 4·23-s + 26-s − 2·28-s − 29-s + 4·31-s − 32-s + 2·34-s + 6·37-s + 2·38-s − 4·41-s + 12·43-s − 4·46-s − 3·49-s − 52-s − 12·53-s + 2·56-s + 58-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 0.196·26-s − 0.377·28-s − 0.185·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.986·37-s + 0.324·38-s − 0.624·41-s + 1.82·43-s − 0.589·46-s − 3/7·49-s − 0.138·52-s − 1.64·53-s + 0.267·56-s + 0.131·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.23336828570210, −12.94967263364839, −12.60237567198292, −11.96536470443285, −11.45139051178469, −11.07833811825410, −10.50466415178528, −10.14924260541252, −9.500521079013722, −9.267201246365294, −8.779444289925276, −8.130033871503825, −7.794209516478673, −7.074671629058241, −6.777519153460337, −6.218166680052892, −5.796832531774865, −5.098978603286474, −4.436312432590247, −4.010695904649966, −3.096191297645616, −2.842621149688672, −2.168160033519263, −1.440376358143113, −0.7028609469363964, 0,
0.7028609469363964, 1.440376358143113, 2.168160033519263, 2.842621149688672, 3.096191297645616, 4.010695904649966, 4.436312432590247, 5.098978603286474, 5.796832531774865, 6.218166680052892, 6.777519153460337, 7.074671629058241, 7.794209516478673, 8.130033871503825, 8.779444289925276, 9.267201246365294, 9.500521079013722, 10.14924260541252, 10.50466415178528, 11.07833811825410, 11.45139051178469, 11.96536470443285, 12.60237567198292, 12.94967263364839, 13.23336828570210
