| L(s) = 1 | + 5·9-s + 4·11-s − 2·19-s + 6·29-s + 8·31-s − 16·41-s + 5·49-s − 2·59-s + 28·61-s + 20·71-s + 20·79-s + 16·81-s + 24·89-s + 20·99-s − 28·101-s − 14·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 1.20·11-s − 0.458·19-s + 1.11·29-s + 1.43·31-s − 2.49·41-s + 5/7·49-s − 0.260·59-s + 3.58·61-s + 2.37·71-s + 2.25·79-s + 16/9·81-s + 2.54·89-s + 2.01·99-s − 2.78·101-s − 1.34·109-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.542076120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.542076120\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.528025274072942944335659489349, −8.380519276914765202130062799307, −7.87812407280192587973044173695, −7.77434848980775936382310046910, −6.88750124009191480934460013894, −6.72798501579055834463631913508, −6.59521731372866378725017474374, −6.53625648618392145621412596375, −5.48266750040847014794515410383, −5.39679155135859262068134042788, −4.82924182779664761793597756632, −4.44301408788742981283387886075, −4.09453531427954223741771908931, −3.74311036894514553072594356463, −3.36063433232076111183966528523, −2.72860112158830787645053082073, −1.94866917095304004562577190739, −1.92830454358131128596931199848, −0.943079307186390359816593907397, −0.824261714270010477597282677940,
0.824261714270010477597282677940, 0.943079307186390359816593907397, 1.92830454358131128596931199848, 1.94866917095304004562577190739, 2.72860112158830787645053082073, 3.36063433232076111183966528523, 3.74311036894514553072594356463, 4.09453531427954223741771908931, 4.44301408788742981283387886075, 4.82924182779664761793597756632, 5.39679155135859262068134042788, 5.48266750040847014794515410383, 6.53625648618392145621412596375, 6.59521731372866378725017474374, 6.72798501579055834463631913508, 6.88750124009191480934460013894, 7.77434848980775936382310046910, 7.87812407280192587973044173695, 8.380519276914765202130062799307, 8.528025274072942944335659489349
