| L(s) = 1 | − 2·3-s − 3·9-s − 6·11-s − 6·17-s + 2·19-s − 25-s + 14·27-s + 12·33-s − 12·41-s − 8·43-s + 12·51-s − 4·57-s − 18·59-s − 14·67-s + 2·73-s + 2·75-s − 4·81-s − 24·83-s − 30·89-s + 20·97-s + 18·99-s + 30·107-s + 12·113-s + 5·121-s + 24·123-s + 127-s + 16·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 1.80·11-s − 1.45·17-s + 0.458·19-s − 1/5·25-s + 2.69·27-s + 2.08·33-s − 1.87·41-s − 1.21·43-s + 1.68·51-s − 0.529·57-s − 2.34·59-s − 1.71·67-s + 0.234·73-s + 0.230·75-s − 4/9·81-s − 2.63·83-s − 3.17·89-s + 2.03·97-s + 1.80·99-s + 2.90·107-s + 1.12·113-s + 5/11·121-s + 2.16·123-s + 0.0887·127-s + 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.898530976505637221100538225737, −7.54540069836880896321995085876, −6.94568874468052569710657332285, −6.40116780242260108785681941586, −6.14873054089518153661475750795, −5.51226753406526396233002934457, −5.28740786687197268416217106929, −4.78385931422370251494541851241, −4.44504131583733523958066512147, −3.37745133212200720528757131098, −2.93923178260800049570600580842, −2.47200231964316304714808376544, −1.56111903440097452457449446667, 0, 0,
1.56111903440097452457449446667, 2.47200231964316304714808376544, 2.93923178260800049570600580842, 3.37745133212200720528757131098, 4.44504131583733523958066512147, 4.78385931422370251494541851241, 5.28740786687197268416217106929, 5.51226753406526396233002934457, 6.14873054089518153661475750795, 6.40116780242260108785681941586, 6.94568874468052569710657332285, 7.54540069836880896321995085876, 7.898530976505637221100538225737
