| L(s) = 1 | + 3-s − 4·7-s + 9-s + 4·11-s − 13-s − 6·17-s − 4·21-s + 4·23-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 2·37-s − 39-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 6·51-s + 2·53-s + 4·59-s + 14·61-s − 4·63-s + 12·67-s + 4·69-s − 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s + 0.520·59-s + 1.79·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835570400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835570400\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.76019256755424498189251499443, −7.01496845223186047092345362595, −6.63453719195890554388236000232, −5.96463827067582150364417721508, −5.00230078087916899658888094512, −3.92383451095228480659126951134, −3.69772392055574236854917436117, −2.68861134755498697986579936170, −1.96347270948825970656484566127, −0.63481869045556656904141253025,
0.63481869045556656904141253025, 1.96347270948825970656484566127, 2.68861134755498697986579936170, 3.69772392055574236854917436117, 3.92383451095228480659126951134, 5.00230078087916899658888094512, 5.96463827067582150364417721508, 6.63453719195890554388236000232, 7.01496845223186047092345362595, 7.76019256755424498189251499443
