| L(s) = 1 | − 3-s − 7-s + 9-s − 6·11-s − 2·13-s − 4·19-s + 21-s + 6·23-s − 27-s + 6·29-s + 8·31-s + 6·33-s − 2·37-s + 2·39-s + 12·41-s + 4·43-s − 12·47-s + 49-s + 6·53-s + 4·57-s − 10·61-s − 63-s − 8·67-s − 6·69-s + 6·71-s + 10·73-s + 6·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.712·71-s + 1.17·73-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9795395896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9795395896\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.160755535964379354523489733441, −8.208792959356633127272389160552, −7.56907459997685032113873998444, −6.68428738506159861567855023989, −5.96686399911741876769524513282, −5.00463732145876448988227255567, −4.53602693910607323330756743987, −3.08167105333254803920115617344, −2.35301068723521734945649298411, −0.64563828008458494707754753873,
0.64563828008458494707754753873, 2.35301068723521734945649298411, 3.08167105333254803920115617344, 4.53602693910607323330756743987, 5.00463732145876448988227255567, 5.96686399911741876769524513282, 6.68428738506159861567855023989, 7.56907459997685032113873998444, 8.208792959356633127272389160552, 9.160755535964379354523489733441
