| L(s) = 1 | + 2·3-s − 2·7-s + 9-s − 4·11-s + 8·17-s + 19-s − 4·21-s − 6·23-s − 4·27-s + 2·29-s + 8·31-s − 8·33-s − 6·41-s − 10·43-s + 6·47-s − 3·49-s + 16·51-s + 2·57-s + 4·59-s + 6·61-s − 2·63-s − 2·67-s − 12·69-s − 16·71-s − 16·73-s + 8·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.94·17-s + 0.229·19-s − 0.872·21-s − 1.25·23-s − 0.769·27-s + 0.371·29-s + 1.43·31-s − 1.39·33-s − 0.937·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s + 2.24·51-s + 0.264·57-s + 0.520·59-s + 0.768·61-s − 0.251·63-s − 0.244·67-s − 1.44·69-s − 1.89·71-s − 1.87·73-s + 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.68561008241802724764661390861, −7.07504432095625409769275078136, −6.02514048906088790000393170014, −5.55567065376411016980907564004, −4.60785047098852732697039519623, −3.60746190908124003957799063681, −3.06821939736646128240408471603, −2.55380355898953321579694218693, −1.43220133260222837339828957141, 0,
1.43220133260222837339828957141, 2.55380355898953321579694218693, 3.06821939736646128240408471603, 3.60746190908124003957799063681, 4.60785047098852732697039519623, 5.55567065376411016980907564004, 6.02514048906088790000393170014, 7.07504432095625409769275078136, 7.68561008241802724764661390861
