| L(s) = 1 | + 2·3-s − 4·7-s + 3·9-s − 8·19-s − 8·21-s − 6·25-s + 4·27-s + 4·29-s + 8·31-s − 4·37-s + 16·47-s + 9·49-s + 20·53-s − 16·57-s − 8·59-s − 12·63-s − 12·75-s + 5·81-s + 24·83-s + 8·87-s + 16·93-s − 24·103-s + 28·109-s − 8·111-s + 4·113-s − 6·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 9-s − 1.83·19-s − 1.74·21-s − 6/5·25-s + 0.769·27-s + 0.742·29-s + 1.43·31-s − 0.657·37-s + 2.33·47-s + 9/7·49-s + 2.74·53-s − 2.11·57-s − 1.04·59-s − 1.51·63-s − 1.38·75-s + 5/9·81-s + 2.63·83-s + 0.857·87-s + 1.65·93-s − 2.36·103-s + 2.68·109-s − 0.759·111-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074609216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074609216\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.743524682466596781725926555582, −8.194996606889196447340137083980, −7.77266654072122543697820690098, −7.08418685931226746330310304373, −6.91469103626909406223255043804, −6.15267013015603292112906972931, −6.11847465912223458443982849872, −5.30972869916546985463135518288, −4.45734868069307623303098179259, −4.10625798014363516984369255368, −3.66065670673094450794263791803, −3.02070160065614261488223955203, −2.44821729992400328434125228202, −2.04703332239703026520690505380, −0.70664651131190107041485094273,
0.70664651131190107041485094273, 2.04703332239703026520690505380, 2.44821729992400328434125228202, 3.02070160065614261488223955203, 3.66065670673094450794263791803, 4.10625798014363516984369255368, 4.45734868069307623303098179259, 5.30972869916546985463135518288, 6.11847465912223458443982849872, 6.15267013015603292112906972931, 6.91469103626909406223255043804, 7.08418685931226746330310304373, 7.77266654072122543697820690098, 8.194996606889196447340137083980, 8.743524682466596781725926555582
