L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·11-s + 4·13-s + 2·15-s − 6·17-s − 8·19-s − 6·23-s − 25-s − 27-s − 10·29-s − 4·31-s − 2·33-s + 6·37-s − 4·39-s + 6·41-s + 4·43-s − 2·45-s − 8·47-s + 6·51-s + 2·53-s − 4·55-s + 8·57-s + 4·59-s + 8·61-s − 8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.348·33-s + 0.986·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 0.840·51-s + 0.274·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.55524886118653010856131298455, −9.274244658387308363441061014553, −8.507433076389310746106303388216, −7.57467808435421420373253219646, −6.50466910249511158423082315406, −5.86682211170824779168620697554, −4.24169896010131124047382143237, −3.93923672216631627490124513607, −1.96358988551248475123570105994, 0,
1.96358988551248475123570105994, 3.93923672216631627490124513607, 4.24169896010131124047382143237, 5.86682211170824779168620697554, 6.50466910249511158423082315406, 7.57467808435421420373253219646, 8.507433076389310746106303388216, 9.274244658387308363441061014553, 10.55524886118653010856131298455