L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s − 11-s + 15-s − 2·17-s + 2·19-s + 21-s − 7·23-s − 4·25-s − 5·27-s + 10·29-s + 7·31-s − 33-s + 35-s + 9·37-s − 2·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s − 2·53-s − 55-s + 2·57-s + 15·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.258·15-s − 0.485·17-s + 0.458·19-s + 0.218·21-s − 1.45·23-s − 4/5·25-s − 0.962·27-s + 1.85·29-s + 1.25·31-s − 0.174·33-s + 0.169·35-s + 1.47·37-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.134·55-s + 0.264·57-s + 1.95·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.477033017\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.477033017\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.164400351857759128500212543992, −7.87404104579587702604894134834, −6.76510153951106529886427450825, −6.04956296563046921215939243259, −5.42136245461813480881489585872, −4.49170145379077500258866260165, −3.73442025194138113182506034259, −2.57891879893810287667817873696, −2.25395729266732590260667429868, −0.830336992343469057674078215670,
0.830336992343469057674078215670, 2.25395729266732590260667429868, 2.57891879893810287667817873696, 3.73442025194138113182506034259, 4.49170145379077500258866260165, 5.42136245461813480881489585872, 6.04956296563046921215939243259, 6.76510153951106529886427450825, 7.87404104579587702604894134834, 8.164400351857759128500212543992