| L(s) = 1 | + 2·11-s − 8·19-s + 10·29-s + 2·31-s + 10·49-s + 8·59-s + 20·61-s + 24·71-s − 30·79-s + 24·89-s − 30·101-s + 8·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 1.83·19-s + 1.85·29-s + 0.359·31-s + 10/7·49-s + 1.04·59-s + 2.56·61-s + 2.84·71-s − 3.37·79-s + 2.54·89-s − 2.98·101-s + 0.766·109-s − 1.72·121-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.924014919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.924014919\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.385505387719458446938286564844, −8.224321525524111180145217636878, −7.67070689557175807334461262451, −7.13515111669526287765585740295, −6.78995086703178448939484440777, −6.72779144001320854807441256832, −6.22505549548480915382681375905, −5.95743199083639168433177707665, −5.41465136428094452442428538847, −5.09962748974859434969231042613, −4.68230215194648102845723579985, −4.20806667609395150143893487289, −3.86943734200286838567029651915, −3.75833817454951547784460863793, −2.85643756423026950406725372067, −2.63368115076741450350968166267, −2.19035269333625169688436513937, −1.66932950422626434459552698595, −0.961687572878066953455925939998, −0.53453983323481518663677549440,
0.53453983323481518663677549440, 0.961687572878066953455925939998, 1.66932950422626434459552698595, 2.19035269333625169688436513937, 2.63368115076741450350968166267, 2.85643756423026950406725372067, 3.75833817454951547784460863793, 3.86943734200286838567029651915, 4.20806667609395150143893487289, 4.68230215194648102845723579985, 5.09962748974859434969231042613, 5.41465136428094452442428538847, 5.95743199083639168433177707665, 6.22505549548480915382681375905, 6.72779144001320854807441256832, 6.78995086703178448939484440777, 7.13515111669526287765585740295, 7.67070689557175807334461262451, 8.224321525524111180145217636878, 8.385505387719458446938286564844
