| L(s) = 1 | − 8·7-s − 20·11-s − 22·13-s − 14·17-s + 76·19-s + 56·23-s + 154·29-s + 160·31-s + 162·37-s + 390·41-s − 388·43-s − 544·47-s − 279·49-s − 210·53-s + 380·59-s − 794·61-s + 148·67-s + 840·71-s − 858·73-s + 160·77-s + 144·79-s + 316·83-s − 1.09e3·89-s + 176·91-s − 994·97-s + 834·101-s − 1.67e3·103-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 0.548·11-s − 0.469·13-s − 0.199·17-s + 0.917·19-s + 0.507·23-s + 0.986·29-s + 0.926·31-s + 0.719·37-s + 1.48·41-s − 1.37·43-s − 1.68·47-s − 0.813·49-s − 0.544·53-s + 0.838·59-s − 1.66·61-s + 0.269·67-s + 1.40·71-s − 1.37·73-s + 0.236·77-s + 0.205·79-s + 0.417·83-s − 1.30·89-s + 0.202·91-s − 1.04·97-s + 0.821·101-s − 1.59·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 154 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 544 T + p^{3} T^{2} \) |
| 53 | \( 1 + 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 794 T + p^{3} T^{2} \) |
| 67 | \( 1 - 148 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 858 T + p^{3} T^{2} \) |
| 79 | \( 1 - 144 T + p^{3} T^{2} \) |
| 83 | \( 1 - 316 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 97 | \( 1 + 994 T + p^{3} T^{2} \) |
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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
−8.414724710604148532055144325730, −7.80073575539387212080742124146, −6.89258712048989335199369530037, −6.19902182744199492882518208934, −5.17229772424063150776086663632, −4.51287273463497839984526854023, −3.25089204399064140921049234847, −2.59715529280120275367682839611, −1.21705030534419392775461473751, 0,
1.21705030534419392775461473751, 2.59715529280120275367682839611, 3.25089204399064140921049234847, 4.51287273463497839984526854023, 5.17229772424063150776086663632, 6.19902182744199492882518208934, 6.89258712048989335199369530037, 7.80073575539387212080742124146, 8.414724710604148532055144325730
