L(s) = 1 | + 3-s + 2·5-s + 9-s − 2·11-s − 13-s + 2·15-s + 19-s − 25-s + 27-s − 4·29-s − 9·31-s − 2·33-s − 3·37-s − 39-s − 10·41-s + 5·43-s + 2·45-s + 6·47-s − 12·53-s − 4·55-s + 57-s − 12·59-s − 10·61-s − 2·65-s − 5·67-s + 6·71-s − 3·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s + 0.229·19-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.61·31-s − 0.348·33-s − 0.493·37-s − 0.160·39-s − 1.56·41-s + 0.762·43-s + 0.298·45-s + 0.875·47-s − 1.64·53-s − 0.539·55-s + 0.132·57-s − 1.56·59-s − 1.28·61-s − 0.248·65-s − 0.610·67-s + 0.712·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55071437538943836623405220284, −6.70438184457472864541167980056, −5.93360966412049476443514792853, −5.34295079124919823511844933757, −4.66125794347334975754976010290, −3.66643801060135496247185274273, −3.00604846405785718442436393902, −2.09629495260824139671739496187, −1.56612496985569481548152977577, 0,
1.56612496985569481548152977577, 2.09629495260824139671739496187, 3.00604846405785718442436393902, 3.66643801060135496247185274273, 4.66125794347334975754976010290, 5.34295079124919823511844933757, 5.93360966412049476443514792853, 6.70438184457472864541167980056, 7.55071437538943836623405220284