| L(s) = 1 | + 2·3-s + 9-s + 11-s + 2·13-s + 2·17-s − 8·19-s − 4·27-s − 6·29-s − 2·31-s + 2·33-s + 10·37-s + 4·39-s − 10·41-s + 8·43-s + 6·47-s + 4·51-s + 6·53-s − 16·57-s + 6·59-s + 2·61-s + 12·67-s − 8·71-s − 2·73-s − 12·79-s − 11·81-s − 8·83-s − 12·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.769·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 1.64·37-s + 0.640·39-s − 1.56·41-s + 1.21·43-s + 0.875·47-s + 0.560·51-s + 0.824·53-s − 2.11·57-s + 0.781·59-s + 0.256·61-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 1.35·79-s − 1.22·81-s − 0.878·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.17484771196450, −13.25093749719328, −13.09419264073828, −12.76719426806057, −11.88445119951795, −11.55105100881498, −10.92762065386521, −10.49015524531309, −9.895405739197119, −9.390266434250487, −8.832806388148104, −8.589422612783461, −8.076152171433475, −7.513791845223085, −7.055817130629392, −6.331939061581131, −5.882355032613533, −5.352123220127354, −4.458170077047114, −3.916109480151259, −3.679682045635122, −2.835384645513297, −2.329521467213436, −1.817187064916911, −0.9970553419345082, 0,
0.9970553419345082, 1.817187064916911, 2.329521467213436, 2.835384645513297, 3.679682045635122, 3.916109480151259, 4.458170077047114, 5.352123220127354, 5.882355032613533, 6.331939061581131, 7.055817130629392, 7.513791845223085, 8.076152171433475, 8.589422612783461, 8.832806388148104, 9.390266434250487, 9.895405739197119, 10.49015524531309, 10.92762065386521, 11.55105100881498, 11.88445119951795, 12.76719426806057, 13.09419264073828, 13.25093749719328, 14.17484771196450
