| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s + 2·17-s − 8·19-s − 4·23-s + 27-s + 6·29-s − 4·33-s + 2·37-s + 2·39-s − 6·41-s + 4·43-s + 12·47-s − 7·49-s + 2·51-s − 6·53-s − 8·57-s − 12·59-s − 14·61-s − 12·67-s − 4·69-s − 2·73-s − 8·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s − 49-s + 0.280·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 1.79·61-s − 1.46·67-s − 0.481·69-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.982491604515988320303136623340, −7.42301803178756849159935957724, −6.35445077174238505678164209387, −5.90567554473049224033311288622, −4.78127660761564881576481451738, −4.21942528211502457767200390993, −3.19308046391431591721660549553, −2.49341630917546362038310153219, −1.54036268239494290990953071664, 0,
1.54036268239494290990953071664, 2.49341630917546362038310153219, 3.19308046391431591721660549553, 4.21942528211502457767200390993, 4.78127660761564881576481451738, 5.90567554473049224033311288622, 6.35445077174238505678164209387, 7.42301803178756849159935957724, 7.982491604515988320303136623340
