| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 4·11-s − 6·13-s − 15-s − 2·17-s + 21-s + 23-s + 25-s + 27-s + 6·29-s − 4·33-s − 35-s + 2·37-s − 6·39-s + 6·41-s − 4·43-s − 45-s + 49-s − 2·51-s + 6·53-s + 4·55-s + 12·59-s + 10·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.97667531621618, −14.67829718095978, −14.14078084691007, −13.51611901167506, −12.97250569362915, −12.58982947691195, −11.95476164614224, −11.49896791280602, −10.83710119510096, −10.19864859951092, −9.941361255289565, −9.216183765015393, −8.533135960429501, −8.167837501058376, −7.559917091659869, −7.163074517981671, −6.607025590479238, −5.588223869230043, −5.107722070682900, −4.538446666951822, −4.032804256269466, −3.040658038359871, −2.562496979686202, −2.101150595855777, −0.9245272055955924, 0,
0.9245272055955924, 2.101150595855777, 2.562496979686202, 3.040658038359871, 4.032804256269466, 4.538446666951822, 5.107722070682900, 5.588223869230043, 6.607025590479238, 7.163074517981671, 7.559917091659869, 8.167837501058376, 8.533135960429501, 9.216183765015393, 9.941361255289565, 10.19864859951092, 10.83710119510096, 11.49896791280602, 11.95476164614224, 12.58982947691195, 12.97250569362915, 13.51611901167506, 14.14078084691007, 14.67829718095978, 14.97667531621618
