L(s) = 1 | + 2-s − 4-s − 2·7-s − 3·8-s + 2·11-s + 2·13-s − 2·14-s − 16-s + 2·17-s + 19-s + 2·22-s + 2·26-s + 2·28-s − 6·29-s − 4·31-s + 5·32-s + 2·34-s − 2·37-s + 38-s + 2·41-s + 10·43-s − 2·44-s − 3·49-s − 2·52-s − 10·53-s + 6·56-s − 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s + 0.603·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.229·19-s + 0.426·22-s + 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s − 0.328·37-s + 0.162·38-s + 0.312·41-s + 1.52·43-s − 0.301·44-s − 3/7·49-s − 0.277·52-s − 1.37·53-s + 0.801·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.985747062585604063106011181204, −7.22117264911175252690776582940, −6.24872892687277686377650206760, −5.86074666577411670109879558457, −5.02859116559444254603579074852, −4.08845500523397408315186958078, −3.56920410441890885951264951499, −2.81096906940086691601576756343, −1.38444333664563009975010701021, 0,
1.38444333664563009975010701021, 2.81096906940086691601576756343, 3.56920410441890885951264951499, 4.08845500523397408315186958078, 5.02859116559444254603579074852, 5.86074666577411670109879558457, 6.24872892687277686377650206760, 7.22117264911175252690776582940, 7.985747062585604063106011181204