| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 6·11-s − 13-s + 4·14-s + 16-s + 6·17-s − 19-s + 6·22-s + 6·23-s + 26-s − 4·28-s + 3·29-s + 2·31-s − 32-s − 6·34-s − 7·37-s + 38-s + 3·41-s − 10·43-s − 6·44-s − 6·46-s + 9·49-s − 52-s − 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 1.27·22-s + 1.25·23-s + 0.196·26-s − 0.755·28-s + 0.557·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 1.15·37-s + 0.162·38-s + 0.468·41-s − 1.52·43-s − 0.904·44-s − 0.884·46-s + 9/7·49-s − 0.138·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5816224499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5816224499\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.99360533116797, −13.47586678865457, −12.83893205873932, −12.70101435625967, −12.17917200749595, −11.50463072010734, −10.84428278596337, −10.40645353081790, −9.948118554788860, −9.724248709398276, −9.080781321707166, −8.337566316907149, −8.079618937591215, −7.378763407756511, −6.924730456525246, −6.460703759501670, −5.736676931283679, −5.260562562284004, −4.772523760683561, −3.656518681890791, −3.109235163320495, −2.850525798249979, −2.084869687614273, −1.086631663142105, −0.3118576028522019,
0.3118576028522019, 1.086631663142105, 2.084869687614273, 2.850525798249979, 3.109235163320495, 3.656518681890791, 4.772523760683561, 5.260562562284004, 5.736676931283679, 6.460703759501670, 6.924730456525246, 7.378763407756511, 8.079618937591215, 8.337566316907149, 9.080781321707166, 9.724248709398276, 9.948118554788860, 10.40645353081790, 10.84428278596337, 11.50463072010734, 12.17917200749595, 12.70101435625967, 12.83893205873932, 13.47586678865457, 13.99360533116797
