| L(s) = 1 | − 4·3-s − 6·5-s − 11·9-s − 12·11-s + 82·13-s + 24·15-s + 30·17-s − 68·19-s + 216·23-s − 89·25-s + 152·27-s + 246·29-s + 112·31-s + 48·33-s + 110·37-s − 328·39-s + 246·41-s − 172·43-s + 66·45-s − 192·47-s − 120·51-s + 558·53-s + 72·55-s + 272·57-s − 540·59-s − 110·61-s − 492·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.536·5-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 + 1.95·23-s − 0.711·25-s + 1.08·27-s + 1.57·29-s + 0.648·31-s + 0.253·33-s + 0.488·37-s − 1.34·39-s + 0.937·41-s − 0.609·43-s + 0.218·45-s − 0.595·47-s − 0.329·51-s + 1.44·53-s + 0.176·55-s + 0.632·57-s − 1.19·59-s − 0.230·61-s − 0.938·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.131336855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131336855\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 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
−11.85071404593962473047836730741, −11.12088321399558509984231927740, −10.43763430242638893479954359749, −8.870979684143155632903234084751, −8.100370882613255034823779324686, −6.66463093464125321589768039413, −5.79107900587177393775136246967, −4.53439395867330685878526981944, −3.10802337710429294921954754920, −0.866367170195998475048506735017,
0.866367170195998475048506735017, 3.10802337710429294921954754920, 4.53439395867330685878526981944, 5.79107900587177393775136246967, 6.66463093464125321589768039413, 8.100370882613255034823779324686, 8.870979684143155632903234084751, 10.43763430242638893479954359749, 11.12088321399558509984231927740, 11.85071404593962473047836730741
