| L(s) = 1 | + 2-s − 4-s − 3·8-s − 2·13-s − 16-s − 2·17-s − 4·19-s − 2·26-s − 2·29-s + 5·32-s − 2·34-s + 10·37-s − 4·38-s + 10·41-s + 4·43-s + 8·47-s − 7·49-s + 2·52-s − 10·53-s − 2·58-s + 4·59-s + 2·61-s + 7·64-s − 12·67-s + 2·68-s + 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.392·26-s − 0.371·29-s + 0.883·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s + 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.277·52-s − 1.37·53-s − 0.262·58-s + 0.520·59-s + 0.256·61-s + 7/8·64-s − 1.46·67-s + 0.242·68-s + 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.53784706591901, −14.70784370564101, −14.51642567340048, −14.01677526147472, −13.28992526700642, −12.82890823414186, −12.61406935511003, −11.89257894432477, −11.26252453717689, −10.81482595420928, −10.05341678007775, −9.390646211217299, −9.149136608245184, −8.377976054993821, −7.821750274991056, −7.195613149222092, −6.291150547443563, −6.041274260239905, −5.274538768684647, −4.547895985452651, −4.278394881285835, −3.529700056681857, −2.696149727714004, −2.179424371995501, −0.9326748827525501, 0,
0.9326748827525501, 2.179424371995501, 2.696149727714004, 3.529700056681857, 4.278394881285835, 4.547895985452651, 5.274538768684647, 6.041274260239905, 6.291150547443563, 7.195613149222092, 7.821750274991056, 8.377976054993821, 9.149136608245184, 9.390646211217299, 10.05341678007775, 10.81482595420928, 11.26252453717689, 11.89257894432477, 12.61406935511003, 12.82890823414186, 13.28992526700642, 14.01677526147472, 14.51642567340048, 14.70784370564101, 15.53784706591901
