| L(s) = 1 | + 2·7-s − 5·9-s + 11-s − 8·13-s − 4·16-s + 4·17-s − 2·23-s − 9·25-s + 6·37-s + 16·41-s − 3·49-s − 12·53-s − 24·61-s − 10·63-s − 14·67-s − 6·71-s − 8·73-s + 2·77-s + 16·81-s + 12·83-s − 16·91-s − 5·99-s − 4·101-s − 8·112-s + 18·113-s + 40·117-s + 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 5/3·9-s + 0.301·11-s − 2.21·13-s − 16-s + 0.970·17-s − 0.417·23-s − 9/5·25-s + 0.986·37-s + 2.49·41-s − 3/7·49-s − 1.64·53-s − 3.07·61-s − 1.25·63-s − 1.71·67-s − 0.712·71-s − 0.936·73-s + 0.227·77-s + 16/9·81-s + 1.31·83-s − 1.67·91-s − 0.502·99-s − 0.398·101-s − 0.755·112-s + 1.69·113-s + 3.69·117-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.410787129321073286385580378997, −9.366222803316740054530803267641, −8.678914216536525529374637909269, −7.78789607043645765017922373874, −7.77729309499855305690824525396, −7.38142851192821190173908719125, −6.23870064789214498217427813253, −5.98958560322778025003846316474, −5.41175071356617719099292784896, −4.56768346065918811492983607197, −4.44574490854809981150981932393, −3.18179592020545258264561636452, −2.63044898935838963010520851422, −1.88740391319498359233714309471, 0,
1.88740391319498359233714309471, 2.63044898935838963010520851422, 3.18179592020545258264561636452, 4.44574490854809981150981932393, 4.56768346065918811492983607197, 5.41175071356617719099292784896, 5.98958560322778025003846316474, 6.23870064789214498217427813253, 7.38142851192821190173908719125, 7.77729309499855305690824525396, 7.78789607043645765017922373874, 8.678914216536525529374637909269, 9.366222803316740054530803267641, 9.410787129321073286385580378997
