| L(s) = 1 | − 3-s + 9-s + 11-s + 4·13-s − 2·17-s − 2·19-s − 8·23-s − 27-s − 6·29-s + 6·31-s − 33-s − 2·37-s − 4·39-s − 6·41-s − 4·43-s − 6·47-s + 2·51-s + 14·53-s + 2·57-s − 8·59-s − 4·61-s − 8·67-s + 8·69-s + 16·71-s + 14·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.485·17-s − 0.458·19-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 1.07·31-s − 0.174·33-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.875·47-s + 0.280·51-s + 1.92·53-s + 0.264·57-s − 1.04·59-s − 0.512·61-s − 0.977·67-s + 0.963·69-s + 1.89·71-s + 1.63·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.78775927772629, −12.20433533950110, −12.03429067058633, −11.45070285186846, −11.00967092010256, −10.72768741058366, −10.04712545812415, −9.823142743227266, −9.205670548339628, −8.616662853698444, −8.306867213393021, −7.833674839174776, −7.211784912234180, −6.610178904716913, −6.308211035612134, −5.964284767952425, −5.274438675012435, −4.913013896279204, −4.056915388532322, −3.963339687610153, −3.350143427549940, −2.555846604909782, −1.866513391184906, −1.526418351104932, −0.6697923844893550, 0,
0.6697923844893550, 1.526418351104932, 1.866513391184906, 2.555846604909782, 3.350143427549940, 3.963339687610153, 4.056915388532322, 4.913013896279204, 5.274438675012435, 5.964284767952425, 6.308211035612134, 6.610178904716913, 7.211784912234180, 7.833674839174776, 8.306867213393021, 8.616662853698444, 9.205670548339628, 9.823142743227266, 10.04712545812415, 10.72768741058366, 11.00967092010256, 11.45070285186846, 12.03429067058633, 12.20433533950110, 12.78775927772629
