L(s) = 1 | + 2·5-s + 7-s − 2·11-s − 2·13-s − 6·17-s − 4·19-s + 6·23-s − 25-s + 4·31-s + 2·35-s − 10·37-s − 2·41-s − 4·43-s + 4·47-s + 49-s − 12·53-s − 4·55-s − 12·59-s − 6·61-s − 4·65-s − 4·67-s − 14·71-s − 2·73-s − 2·77-s + 8·79-s + 16·83-s − 12·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.718·31-s + 0.338·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s − 0.539·55-s − 1.56·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 1.66·71-s − 0.234·73-s − 0.227·77-s + 0.900·79-s + 1.75·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.141634081829378081441809236752, −7.28267859617118153010735912842, −6.56064108292960750609847244365, −5.92082892086786754586742253811, −4.89364263062340315050988534287, −4.61650350267300135260926956930, −3.24669297328856350146221556739, −2.33542067219372894622175613585, −1.65405632235840502550099569734, 0,
1.65405632235840502550099569734, 2.33542067219372894622175613585, 3.24669297328856350146221556739, 4.61650350267300135260926956930, 4.89364263062340315050988534287, 5.92082892086786754586742253811, 6.56064108292960750609847244365, 7.28267859617118153010735912842, 8.141634081829378081441809236752