| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 18·5-s + 6·6-s + 8·8-s + 9·9-s − 36·10-s − 72·11-s + 12·12-s + 34·13-s − 54·15-s + 16·16-s − 6·17-s + 18·18-s − 92·19-s − 72·20-s − 144·22-s − 180·23-s + 24·24-s + 199·25-s + 68·26-s + 27·27-s − 114·29-s − 108·30-s − 56·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.60·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.13·10-s − 1.97·11-s + 0.288·12-s + 0.725·13-s − 0.929·15-s + 1/4·16-s − 0.0856·17-s + 0.235·18-s − 1.11·19-s − 0.804·20-s − 1.39·22-s − 1.63·23-s + 0.204·24-s + 1.59·25-s + 0.512·26-s + 0.192·27-s − 0.729·29-s − 0.657·30-s − 0.324·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 - 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 - 56 T + p^{3} T^{2} \) |
| 83 | \( 1 + 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−10.97474816737360486162337626119, −10.25317105050435823981548610258, −8.509498947699219846616304764988, −7.975447365424881548574785223267, −7.15737886775011953916029096960, −5.68436256236064130887913325054, −4.37771440110268506987800168688, −3.61478561851197937848687506948, −2.37335539335967434378003044451, 0,
2.37335539335967434378003044451, 3.61478561851197937848687506948, 4.37771440110268506987800168688, 5.68436256236064130887913325054, 7.15737886775011953916029096960, 7.975447365424881548574785223267, 8.509498947699219846616304764988, 10.25317105050435823981548610258, 10.97474816737360486162337626119
