L(s) = 1 | + 3-s + 5-s + 9-s + 2·11-s + 13-s + 15-s − 17-s − 8·19-s − 6·23-s + 25-s + 27-s + 4·31-s + 2·33-s − 2·37-s + 39-s + 4·41-s − 4·43-s + 45-s − 7·49-s − 51-s + 6·53-s + 2·55-s − 8·57-s + 4·59-s − 8·61-s + 65-s + 8·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.718·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s + 0.149·45-s − 49-s − 0.140·51-s + 0.824·53-s + 0.269·55-s − 1.05·57-s + 0.520·59-s − 1.02·61-s + 0.124·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.21458644711248, −12.84374824679562, −12.44901958988004, −11.78722846478709, −11.46892639286500, −10.79802097950687, −10.28640847668498, −10.08275859842730, −9.397610186921991, −8.955228253980079, −8.586969403731155, −8.078317436252829, −7.694221110343015, −6.901564698271123, −6.411502017875825, −6.265592410881305, −5.549433203184704, −4.864344788967873, −4.303647037838693, −3.939879610472496, −3.360152306804250, −2.628722938942695, −2.071150437672775, −1.727105794836422, −0.8828983440055012, 0,
0.8828983440055012, 1.727105794836422, 2.071150437672775, 2.628722938942695, 3.360152306804250, 3.939879610472496, 4.303647037838693, 4.864344788967873, 5.549433203184704, 6.265592410881305, 6.411502017875825, 6.901564698271123, 7.694221110343015, 8.078317436252829, 8.586969403731155, 8.955228253980079, 9.397610186921991, 10.08275859842730, 10.28640847668498, 10.79802097950687, 11.46892639286500, 11.78722846478709, 12.44901958988004, 12.84374824679562, 13.21458644711248