| L(s) = 1 | − 6·5-s − 7-s − 3·9-s − 4·16-s + 8·17-s + 17·25-s + 6·35-s − 8·37-s − 12·41-s − 2·43-s + 18·45-s + 14·47-s + 49-s + 16·59-s + 3·63-s − 12·67-s + 6·79-s + 24·80-s + 9·81-s + 30·83-s − 48·85-s + 6·89-s − 28·101-s − 4·109-s + 4·112-s − 8·119-s + 14·121-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 0.377·7-s − 9-s − 16-s + 1.94·17-s + 17/5·25-s + 1.01·35-s − 1.31·37-s − 1.87·41-s − 0.304·43-s + 2.68·45-s + 2.04·47-s + 1/7·49-s + 2.08·59-s + 0.377·63-s − 1.46·67-s + 0.675·79-s + 2.68·80-s + 81-s + 3.29·83-s − 5.20·85-s + 0.635·89-s − 2.78·101-s − 0.383·109-s + 0.377·112-s − 0.733·119-s + 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521703 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521703 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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
−8.183647839794538377077138501226, −7.88305646625584781075065151406, −7.43563929744309328324545511818, −7.03653425752268447358801700101, −6.65228819509679766629569332232, −5.93341763181382776531686272081, −5.27883313770925671699270459709, −5.04373634162543047152508893884, −4.19748278425106429037453036388, −3.85909330468379534544131134495, −3.34023383497486740159428280726, −3.11744276397134147631028514628, −2.15138597706296064169238441165, −0.796423925859051403693435508428, 0,
0.796423925859051403693435508428, 2.15138597706296064169238441165, 3.11744276397134147631028514628, 3.34023383497486740159428280726, 3.85909330468379534544131134495, 4.19748278425106429037453036388, 5.04373634162543047152508893884, 5.27883313770925671699270459709, 5.93341763181382776531686272081, 6.65228819509679766629569332232, 7.03653425752268447358801700101, 7.43563929744309328324545511818, 7.88305646625584781075065151406, 8.183647839794538377077138501226
