L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s + 11-s − 6·13-s + 2·15-s + 2·17-s − 4·19-s + 4·21-s − 4·23-s − 25-s + 27-s + 6·29-s + 33-s + 8·35-s + 6·37-s − 6·39-s − 6·41-s − 4·43-s + 2·45-s + 12·47-s + 9·49-s + 2·51-s + 2·53-s + 2·55-s − 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s + 1.35·35-s + 0.986·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s + 0.269·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.194059710\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.194059710\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.63457399256275619690777187392, −9.964740834749841047239774990320, −9.101258160534375304185395204218, −8.126100348919150852159091302339, −7.47499472452228970996968762649, −6.22756869331455955650272528651, −5.08660020809670672027476290340, −4.30977538214279742560651831455, −2.56194464227123448811095009052, −1.69573254513794375477006333754,
1.69573254513794375477006333754, 2.56194464227123448811095009052, 4.30977538214279742560651831455, 5.08660020809670672027476290340, 6.22756869331455955650272528651, 7.47499472452228970996968762649, 8.126100348919150852159091302339, 9.101258160534375304185395204218, 9.964740834749841047239774990320, 10.63457399256275619690777187392