| L(s) = 1 | + 3-s − 5-s + 9-s + 2·13-s − 15-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 2·29-s + 31-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s − 7·49-s + 6·51-s + 10·53-s + 4·57-s + 12·59-s − 10·61-s − 2·65-s + 4·67-s − 4·69-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.179·31-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.481·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.991773766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.991773766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.01253576399936, −14.68934302541223, −14.11131747659472, −13.54421732564357, −13.21394556435156, −12.39687059239036, −11.90050079747187, −11.64611003080993, −10.71500930236369, −10.34568710200721, −9.537165270596482, −9.395860425017619, −8.371638156938433, −8.088602778787462, −7.636376930622977, −6.908151603633745, −6.348722181533343, −5.481563492473888, −5.117656210614012, −4.153399583000393, −3.567068723467665, −3.211997287139536, −2.286790944996329, −1.454759544044686, −0.6787475454761288,
0.6787475454761288, 1.454759544044686, 2.286790944996329, 3.211997287139536, 3.567068723467665, 4.153399583000393, 5.117656210614012, 5.481563492473888, 6.348722181533343, 6.908151603633745, 7.636376930622977, 8.088602778787462, 8.371638156938433, 9.395860425017619, 9.537165270596482, 10.34568710200721, 10.71500930236369, 11.64611003080993, 11.90050079747187, 12.39687059239036, 13.21394556435156, 13.54421732564357, 14.11131747659472, 14.68934302541223, 15.01253576399936
