| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s + 15-s − 17-s + 8·23-s + 25-s + 27-s − 6·29-s − 4·33-s + 2·37-s − 6·41-s + 8·43-s + 45-s − 7·49-s − 51-s − 6·53-s − 4·55-s − 4·59-s − 10·61-s + 4·67-s + 8·69-s + 14·73-s + 75-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.258·15-s − 0.242·17-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.328·37-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 49-s − 0.140·51-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.488·67-s + 0.963·69-s + 1.63·73-s + 0.115·75-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.83648677496913, −12.52119085054045, −12.00726278758498, −11.15094461753918, −10.92880585482605, −10.68801126361780, −9.952068800933539, −9.474314341938261, −9.290943306278581, −8.699987532782059, −8.148742106504306, −7.796483943592227, −7.329375480652914, −6.792698180611253, −6.361228854342425, −5.700342814326653, −5.195822382627947, −4.888545107364701, −4.281453159444861, −3.595016096381362, −3.046659486667520, −2.707060623156964, −2.064203540367662, −1.565814200376234, −0.8030975442160058, 0,
0.8030975442160058, 1.565814200376234, 2.064203540367662, 2.707060623156964, 3.046659486667520, 3.595016096381362, 4.281453159444861, 4.888545107364701, 5.195822382627947, 5.700342814326653, 6.361228854342425, 6.792698180611253, 7.329375480652914, 7.796483943592227, 8.148742106504306, 8.699987532782059, 9.290943306278581, 9.474314341938261, 9.952068800933539, 10.68801126361780, 10.92880585482605, 11.15094461753918, 12.00726278758498, 12.52119085054045, 12.83648677496913
