| L(s) = 1 | + 4·3-s + 6·5-s + 7·7-s − 11·9-s − 12·11-s − 82·13-s + 24·15-s − 30·17-s + 68·19-s + 28·21-s + 216·23-s − 89·25-s − 152·27-s + 246·29-s − 112·31-s − 48·33-s + 42·35-s + 110·37-s − 328·39-s − 246·41-s − 172·43-s − 66·45-s + 192·47-s + 49·49-s − 120·51-s + 558·53-s − 72·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.536·5-s + 0.377·7-s − 0.407·9-s − 0.328·11-s − 1.74·13-s + 0.413·15-s − 0.428·17-s + 0.821·19-s + 0.290·21-s + 1.95·23-s − 0.711·25-s − 1.08·27-s + 1.57·29-s − 0.648·31-s − 0.253·33-s + 0.202·35-s + 0.488·37-s − 1.34·39-s − 0.937·41-s − 0.609·43-s − 0.218·45-s + 0.595·47-s + 1/7·49-s − 0.329·51-s + 1.44·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.458797723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458797723\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 216 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 550 T + p^{3} T^{2} \) |
| 79 | \( 1 + 208 T + p^{3} T^{2} \) |
| 83 | \( 1 - 516 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1586 T + p^{3} T^{2} \) |
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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
−16.91464440552115059611398230166, −15.20398885950737626094535820850, −14.32945295367607849958636091763, −13.25316910488295587286964041636, −11.70635812900203603357626649826, −10.02923638757392404020866152734, −8.799583145182551892031258097314, −7.31312267248270248202945892893, −5.15882247099216722895824219301, −2.64008443943042920582771265063,
2.64008443943042920582771265063, 5.15882247099216722895824219301, 7.31312267248270248202945892893, 8.799583145182551892031258097314, 10.02923638757392404020866152734, 11.70635812900203603357626649826, 13.25316910488295587286964041636, 14.32945295367607849958636091763, 15.20398885950737626094535820850, 16.91464440552115059611398230166
