L(s) = 1 | + 3·3-s + 6·9-s − 2·11-s + 6·13-s − 2·17-s + 9·23-s + 9·27-s + 3·29-s + 2·31-s − 6·33-s − 8·37-s + 18·39-s + 5·41-s − 43-s − 8·47-s − 6·51-s − 4·53-s − 8·59-s + 7·61-s + 3·67-s + 27·69-s + 8·71-s − 14·73-s + 4·79-s + 9·81-s + 83-s + 9·87-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2·9-s − 0.603·11-s + 1.66·13-s − 0.485·17-s + 1.87·23-s + 1.73·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s + 2.88·39-s + 0.780·41-s − 0.152·43-s − 1.16·47-s − 0.840·51-s − 0.549·53-s − 1.04·59-s + 0.896·61-s + 0.366·67-s + 3.25·69-s + 0.949·71-s − 1.63·73-s + 0.450·79-s + 81-s + 0.109·83-s + 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.224622883\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.224622883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.380131631963893025206910065754, −7.76758445182270513587559873397, −6.96279520232309118377592529104, −6.32114071811963696159466261957, −5.15879786084617746332057457578, −4.36947048291252831226319173401, −3.42377252052257186340198590789, −3.04556749939185228285745615815, −2.05972265777085223325251219548, −1.12413039392067516368134337536,
1.12413039392067516368134337536, 2.05972265777085223325251219548, 3.04556749939185228285745615815, 3.42377252052257186340198590789, 4.36947048291252831226319173401, 5.15879786084617746332057457578, 6.32114071811963696159466261957, 6.96279520232309118377592529104, 7.76758445182270513587559873397, 8.380131631963893025206910065754