| L(s) = 1 | − 9-s − 8·11-s − 8·19-s − 4·29-s − 8·31-s + 4·41-s − 2·49-s − 8·59-s + 12·61-s + 32·71-s + 8·79-s + 81-s − 20·89-s + 8·99-s − 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s − 1.83·19-s − 0.742·29-s − 1.43·31-s + 0.624·41-s − 2/7·49-s − 1.04·59-s + 1.53·61-s + 3.79·71-s + 0.900·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s − 1.19·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.758013295792964580189445896610, −8.175334856283479565184366458135, −8.031608597980090646472922657602, −7.86098078024792670901306255406, −7.18429845183093739347714495190, −6.98253808718326664738695357366, −6.25933822836416351247379396935, −6.20534235550408897063443770371, −5.36560366892489293594598388830, −5.28330711136303118673229284591, −5.05766962892158957405227478660, −4.31489484676743874612318866459, −3.82744557524133953554722235651, −3.56822262297053512783743584258, −2.69904648162715290918230431263, −2.41997882473594672394905060754, −2.16721683340337093038419781585, −1.27153017758547989564993590404, 0, 0,
1.27153017758547989564993590404, 2.16721683340337093038419781585, 2.41997882473594672394905060754, 2.69904648162715290918230431263, 3.56822262297053512783743584258, 3.82744557524133953554722235651, 4.31489484676743874612318866459, 5.05766962892158957405227478660, 5.28330711136303118673229284591, 5.36560366892489293594598388830, 6.20534235550408897063443770371, 6.25933822836416351247379396935, 6.98253808718326664738695357366, 7.18429845183093739347714495190, 7.86098078024792670901306255406, 8.031608597980090646472922657602, 8.175334856283479565184366458135, 8.758013295792964580189445896610
