| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 12-s + 13-s + 14-s − 15-s + 16-s − 4·17-s − 18-s − 4·19-s + 20-s + 21-s + 23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s + 29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.65365656041717, −13.40540652015609, −12.80852873693235, −12.30270516062754, −11.82731915607687, −11.32242944797928, −10.88636084013231, −10.33324004444666, −9.987100362536118, −9.562959558399279, −8.904514856019011, −8.385327272772315, −8.176749457408279, −7.085206146092807, −6.972054368934472, −6.340110011701636, −5.953896753081534, −5.401823967300552, −4.656197704837293, −4.176147517427489, −3.481540812682204, −2.677195041773116, −2.168705237335655, −1.525697887894281, −0.7247359278412741, 0,
0.7247359278412741, 1.525697887894281, 2.168705237335655, 2.677195041773116, 3.481540812682204, 4.176147517427489, 4.656197704837293, 5.401823967300552, 5.953896753081534, 6.340110011701636, 6.972054368934472, 7.085206146092807, 8.176749457408279, 8.385327272772315, 8.904514856019011, 9.562959558399279, 9.987100362536118, 10.33324004444666, 10.88636084013231, 11.32242944797928, 11.82731915607687, 12.30270516062754, 12.80852873693235, 13.40540652015609, 13.65365656041717
