| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s − 14-s − 15-s + 16-s + 18-s + 6·19-s + 20-s + 21-s − 24-s + 25-s + 2·26-s − 27-s − 28-s − 4·29-s − 30-s + 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.742·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.897565592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.897565592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.49282617699283, −12.02493889329629, −11.70702468969067, −11.17005491870143, −10.85010544309820, −10.25394675781414, −9.837017447380533, −9.513367144604602, −8.916514799279801, −8.322286608209294, −7.836704834074423, −7.257822184346051, −6.812704584165955, −6.415509522411630, −5.945855617010109, −5.356031772037938, −5.237772520563437, −4.563811476869611, −3.850828033798320, −3.631299176507940, −2.879046135582783, −2.447410320847685, −1.714970633824701, −1.093446800326998, −0.5725908968069788,
0.5725908968069788, 1.093446800326998, 1.714970633824701, 2.447410320847685, 2.879046135582783, 3.631299176507940, 3.850828033798320, 4.563811476869611, 5.237772520563437, 5.356031772037938, 5.945855617010109, 6.415509522411630, 6.812704584165955, 7.257822184346051, 7.836704834074423, 8.322286608209294, 8.916514799279801, 9.513367144604602, 9.837017447380533, 10.25394675781414, 10.85010544309820, 11.17005491870143, 11.70702468969067, 12.02493889329629, 12.49282617699283
