| L(s) = 1 | − 11·7-s − 23·13-s − 74·19-s − 25·25-s + 46·31-s − 146·37-s + 22·43-s + 49·49-s − 47·61-s + 13·67-s + 286·73-s − 11·79-s + 253·91-s + 169·97-s + 157·103-s − 428·109-s − 121·121-s + 127-s + 131-s + 814·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.57·7-s − 1.76·13-s − 3.89·19-s − 25-s + 1.48·31-s − 3.94·37-s + 0.511·43-s + 49-s − 0.770·61-s + 0.194·67-s + 3.91·73-s − 0.139·79-s + 2.78·91-s + 1.74·97-s + 1.52·103-s − 3.92·109-s − 121-s + 0.00787·127-s + 0.00763·131-s + 6.12·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09146329456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09146329456\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19787016526518731289295178227, −10.80692979765177978195100061491, −10.75836997100981482954025134063, −10.07161338009180343209276048616, −10.01742406937010054516506518100, −9.240742061622227408066858555585, −8.956745681547651525506563841765, −8.242177038758144632334732255818, −8.029560461999959198162040234432, −6.92837239049983512834012919640, −6.89405383267080586217379509987, −6.38421989711552048256758718644, −5.89166597756973500113469119646, −5.04188097132166076130779560552, −4.57565952732927872104641596823, −3.87126525386735011037495887685, −3.35090367058798269526636706664, −2.28677759127080326458757997554, −2.15353844291880143779285500069, −0.13090030908602776820523889904,
0.13090030908602776820523889904, 2.15353844291880143779285500069, 2.28677759127080326458757997554, 3.35090367058798269526636706664, 3.87126525386735011037495887685, 4.57565952732927872104641596823, 5.04188097132166076130779560552, 5.89166597756973500113469119646, 6.38421989711552048256758718644, 6.89405383267080586217379509987, 6.92837239049983512834012919640, 8.029560461999959198162040234432, 8.242177038758144632334732255818, 8.956745681547651525506563841765, 9.240742061622227408066858555585, 10.01742406937010054516506518100, 10.07161338009180343209276048616, 10.75836997100981482954025134063, 10.80692979765177978195100061491, 12.19787016526518731289295178227
