| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 2·11-s − 2·13-s + 16-s + 2·17-s − 8·19-s + 20-s + 2·22-s + 4·23-s + 25-s + 2·26-s + 10·29-s + 2·31-s − 32-s − 2·34-s + 37-s + 8·38-s − 40-s − 8·41-s − 6·43-s − 2·44-s − 4·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 1.83·19-s + 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.85·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.164·37-s + 1.29·38-s − 0.158·40-s − 1.24·41-s − 0.914·43-s − 0.301·44-s − 0.589·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.408143730107138610044692248639, −7.64963676724052456719193587557, −6.68601144599048155922943535024, −6.30349379090631139189278339381, −5.17339540004283311638824595430, −4.55720531529590455934609038280, −3.18386506034984626649901490212, −2.44876920041597401980141388769, −1.42856017407325710747629050042, 0,
1.42856017407325710747629050042, 2.44876920041597401980141388769, 3.18386506034984626649901490212, 4.55720531529590455934609038280, 5.17339540004283311638824595430, 6.30349379090631139189278339381, 6.68601144599048155922943535024, 7.64963676724052456719193587557, 8.408143730107138610044692248639
