| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 4·13-s − 15-s + 6·17-s − 2·19-s + 4·21-s + 25-s + 27-s − 4·31-s − 4·35-s − 10·37-s + 4·39-s + 4·43-s − 45-s + 12·47-s + 9·49-s + 6·51-s + 6·53-s − 2·57-s + 12·59-s + 10·61-s + 4·63-s − 4·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s + 1.45·17-s − 0.458·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.676·35-s − 1.64·37-s + 0.640·39-s + 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s + 1.56·59-s + 1.28·61-s + 0.503·63-s − 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.338937952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.338937952\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.920197905887042108477183329589, −7.46886618912998610853901995461, −6.69829210899175708120148320877, −5.54377642910380454537508051951, −5.23191560199988675466329056227, −4.00717671958941120920055803380, −3.83521687721507734475596725377, −2.68688181817281924611510903549, −1.72074996519328214405645526453, −0.984342973712545945520052821545,
0.984342973712545945520052821545, 1.72074996519328214405645526453, 2.68688181817281924611510903549, 3.83521687721507734475596725377, 4.00717671958941120920055803380, 5.23191560199988675466329056227, 5.54377642910380454537508051951, 6.69829210899175708120148320877, 7.46886618912998610853901995461, 7.920197905887042108477183329589
