L(s) = 1 | − 3-s − 2·4-s − 5-s + 9-s − 11-s + 2·12-s + 4·13-s + 15-s + 4·16-s + 5·17-s − 19-s + 2·20-s − 5·23-s + 25-s − 27-s + 3·29-s + 6·31-s + 33-s − 2·36-s − 12·37-s − 4·39-s + 2·41-s + 13·43-s + 2·44-s − 45-s + 6·47-s − 4·48-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 1.10·13-s + 0.258·15-s + 16-s + 1.21·17-s − 0.229·19-s + 0.447·20-s − 1.04·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s + 0.174·33-s − 1/3·36-s − 1.97·37-s − 0.640·39-s + 0.312·41-s + 1.98·43-s + 0.301·44-s − 0.149·45-s + 0.875·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.067938853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.067938853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.84723442237906906820094973225, −7.27750757370309344841929065322, −6.20389316522814694487118088538, −5.74489622162344058924311163230, −5.06001105852376010052631087747, −4.19726227029648487768249831067, −3.78935351762163263406767062403, −2.85101923914972999085304036840, −1.41556512842197637507997614218, −0.58436025172717371070558099849,
0.58436025172717371070558099849, 1.41556512842197637507997614218, 2.85101923914972999085304036840, 3.78935351762163263406767062403, 4.19726227029648487768249831067, 5.06001105852376010052631087747, 5.74489622162344058924311163230, 6.20389316522814694487118088538, 7.27750757370309344841929065322, 7.84723442237906906820094973225