L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·13-s + 17-s + 8·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s − 6·29-s + 4·31-s + 2·37-s − 4·39-s + 6·41-s − 4·43-s − 6·47-s − 3·49-s − 51-s − 12·53-s − 8·57-s + 4·61-s + 2·63-s + 4·67-s + 6·69-s − 6·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.242·17-s + 1.83·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s − 0.140·51-s − 1.64·53-s − 1.05·57-s + 0.512·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.165124912\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.165124912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.78154800711207, −13.29196197354228, −12.84048913752751, −12.13853764204830, −11.64187296998390, −11.39831974462283, −11.07653208391269, −10.19150531208318, −9.970898860269355, −9.369673897763165, −8.830730950992124, −8.092794666237780, −7.648226690122762, −7.527624162372386, −6.388378860955549, −6.205993093026370, −5.604325172858189, −5.049857881107896, −4.601348119941479, −3.686977163154519, −3.575386881689912, −2.581111974310106, −1.698558063061971, −1.362849893853326, −0.5003615018639221,
0.5003615018639221, 1.362849893853326, 1.698558063061971, 2.581111974310106, 3.575386881689912, 3.686977163154519, 4.601348119941479, 5.049857881107896, 5.604325172858189, 6.205993093026370, 6.388378860955549, 7.527624162372386, 7.648226690122762, 8.092794666237780, 8.830730950992124, 9.369673897763165, 9.970898860269355, 10.19150531208318, 11.07653208391269, 11.39831974462283, 11.64187296998390, 12.13853764204830, 12.84048913752751, 13.29196197354228, 13.78154800711207