| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 4·11-s − 2·12-s + 2·13-s + 16-s − 8·17-s − 18-s − 5·19-s − 4·22-s − 23-s + 2·24-s − 2·26-s + 4·27-s + 10·29-s − 4·31-s − 32-s − 8·33-s + 8·34-s + 36-s − 37-s + 5·38-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.852·22-s − 0.208·23-s + 0.408·24-s − 0.392·26-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 1.39·33-s + 1.37·34-s + 1/6·36-s − 0.164·37-s + 0.811·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.832914826819812321147395721742, −8.313688419694499349085941483242, −6.99952647899610143196985263278, −6.41576483049882713415932622776, −6.04441513309007745635467932400, −4.73812112133878014134329736510, −4.03316263506278759152895715932, −2.53069878250121404567366643784, −1.29600857288238140755146691958, 0,
1.29600857288238140755146691958, 2.53069878250121404567366643784, 4.03316263506278759152895715932, 4.73812112133878014134329736510, 6.04441513309007745635467932400, 6.41576483049882713415932622776, 6.99952647899610143196985263278, 8.313688419694499349085941483242, 8.832914826819812321147395721742
