| L(s) = 1 | + 2·5-s − 7-s − 2·13-s − 2·17-s − 4·19-s − 25-s + 6·29-s − 2·35-s − 6·37-s + 6·41-s − 8·43-s − 8·47-s + 49-s + 6·53-s − 12·59-s − 10·61-s − 4·65-s − 16·67-s + 8·71-s − 6·73-s + 8·79-s − 12·83-s − 4·85-s + 14·89-s + 2·91-s − 8·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s − 0.338·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 1.95·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s − 1.31·83-s − 0.433·85-s + 1.48·89-s + 0.209·91-s − 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.135182102936898960408555490697, −7.26276202485153591232285109227, −6.44136834602638133605662170877, −6.04844318255358604898058696904, −5.04291005577214527278511537505, −4.40417236841009185031547150853, −3.27919214186183713470671849023, −2.41533067588935176938476948121, −1.58050365903874040036738648185, 0,
1.58050365903874040036738648185, 2.41533067588935176938476948121, 3.27919214186183713470671849023, 4.40417236841009185031547150853, 5.04291005577214527278511537505, 6.04844318255358604898058696904, 6.44136834602638133605662170877, 7.26276202485153591232285109227, 8.135182102936898960408555490697
