L(s) = 1 | − 3-s + 5-s + 9-s − 5·11-s − 13-s − 15-s + 17-s + 5·19-s + 23-s − 4·25-s − 27-s + 2·29-s − 2·31-s + 5·33-s − 4·37-s + 39-s + 5·41-s + 43-s + 45-s − 51-s − 12·53-s − 5·55-s − 5·57-s + 10·59-s − 10·61-s − 65-s − 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.14·19-s + 0.208·23-s − 4/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.870·33-s − 0.657·37-s + 0.160·39-s + 0.780·41-s + 0.152·43-s + 0.149·45-s − 0.140·51-s − 1.64·53-s − 0.674·55-s − 0.662·57-s + 1.30·59-s − 1.28·61-s − 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9483053514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9483053514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.21985915514682, −12.83535458630900, −12.35507766755856, −11.81969161737321, −11.44011375693933, −10.77680965543479, −10.49486382408557, −9.933980228727790, −9.630496188290986, −9.050028059429172, −8.407717671750246, −7.819693117240686, −7.415760987507621, −7.092890152197311, −6.212045872505873, −5.828090264401637, −5.437835982352758, −4.874507094235602, −4.513159415813273, −3.627362939867628, −3.042191051765857, −2.547329466639430, −1.817854404901042, −1.197600838368884, −0.3032649939871063,
0.3032649939871063, 1.197600838368884, 1.817854404901042, 2.547329466639430, 3.042191051765857, 3.627362939867628, 4.513159415813273, 4.874507094235602, 5.437835982352758, 5.828090264401637, 6.212045872505873, 7.092890152197311, 7.415760987507621, 7.819693117240686, 8.407717671750246, 9.050028059429172, 9.630496188290986, 9.933980228727790, 10.49486382408557, 10.77680965543479, 11.44011375693933, 11.81969161737321, 12.35507766755856, 12.83535458630900, 13.21985915514682