| L(s) = 1 | − 1.87·3-s + 4-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 16-s − 19-s + 25-s − 2.87·27-s + 2.53·36-s − 0.652·39-s + 0.347·47-s − 1.87·48-s + 49-s + 0.347·52-s − 53-s + 1.87·57-s + 1.53·59-s + 64-s + 1.53·73-s − 1.87·75-s − 76-s + 1.53·79-s + 2.87·81-s − 89-s + 100-s + 0.347·101-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 4-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 16-s − 19-s + 25-s − 2.87·27-s + 2.53·36-s − 0.652·39-s + 0.347·47-s − 1.87·48-s + 49-s + 0.347·52-s − 53-s + 1.87·57-s + 1.53·59-s + 64-s + 1.53·73-s − 1.87·75-s − 76-s + 1.53·79-s + 2.87·81-s − 89-s + 100-s + 0.347·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8359489183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8359489183\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2003 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.87T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.347T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.347T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.674909252977817404995822253806, −8.459869010827793059582913584629, −7.41712167701710131552400132536, −6.74031035280392642966854844786, −6.24649343785626698549659674428, −5.52519585731075404240487209219, −4.71935729270348960319489264639, −3.69986029787874061932208308603, −2.21453177929666434446917857794, −1.03009713626109061612792543694,
1.03009713626109061612792543694, 2.21453177929666434446917857794, 3.69986029787874061932208308603, 4.71935729270348960319489264639, 5.52519585731075404240487209219, 6.24649343785626698549659674428, 6.74031035280392642966854844786, 7.41712167701710131552400132536, 8.459869010827793059582913584629, 9.674909252977817404995822253806
