| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s + 13-s − 4·14-s + 16-s + 6·19-s + 20-s + 4·23-s + 25-s + 26-s − 4·28-s + 10·29-s − 8·31-s + 32-s − 4·35-s + 12·37-s + 6·38-s + 40-s + 2·41-s + 6·43-s + 4·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.676·35-s + 1.97·37-s + 0.973·38-s + 0.158·40-s + 0.312·41-s + 0.914·43-s + 0.589·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.84792220211311, −12.44252840417349, −12.04102332850028, −11.58406440909263, −10.84307658890487, −10.72602424823664, −10.01150701072205, −9.647913353792613, −9.279030255842466, −8.824781238830631, −8.217159067414658, −7.429001977000811, −7.196971508561688, −6.759349546705449, −6.118204911002112, −5.705476004868254, −5.606103353089061, −4.613123957311700, −4.365937279795326, −3.646756410075463, −3.112200856024654, −2.771031239722113, −2.379975017959339, −1.233849502656418, −1.019390576703798, 0,
1.019390576703798, 1.233849502656418, 2.379975017959339, 2.771031239722113, 3.112200856024654, 3.646756410075463, 4.365937279795326, 4.613123957311700, 5.606103353089061, 5.705476004868254, 6.118204911002112, 6.759349546705449, 7.196971508561688, 7.429001977000811, 8.217159067414658, 8.824781238830631, 9.279030255842466, 9.647913353792613, 10.01150701072205, 10.72602424823664, 10.84307658890487, 11.58406440909263, 12.04102332850028, 12.44252840417349, 12.84792220211311
