| L(s) = 1 | − 3-s − 3·5-s + 2·7-s − 2·9-s − 13-s + 3·15-s + 4·19-s − 2·21-s − 6·23-s + 4·25-s + 5·27-s + 7·29-s + 8·31-s − 6·35-s − 37-s + 39-s − 8·41-s + 5·43-s + 6·45-s + 47-s − 3·49-s − 12·53-s − 4·57-s − 15·61-s − 4·63-s + 3·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s − 0.277·13-s + 0.774·15-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 4/5·25-s + 0.962·27-s + 1.29·29-s + 1.43·31-s − 1.01·35-s − 0.164·37-s + 0.160·39-s − 1.24·41-s + 0.762·43-s + 0.894·45-s + 0.145·47-s − 3/7·49-s − 1.64·53-s − 0.529·57-s − 1.92·61-s − 0.503·63-s + 0.372·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−15.66273738058336, −15.13087674433756, −14.25905173338017, −14.14647620775913, −13.53563059133528, −12.46596255244797, −12.15219886196529, −11.75504584873869, −11.38514833358504, −10.84079068339231, −10.20374865864168, −9.650047934354370, −8.694870801794323, −8.307744087690806, −7.839430472800992, −7.369839623189211, −6.520373048176753, −6.044073908800863, −5.241342675399931, −4.637228090350433, −4.318982230269526, −3.284443241327431, −2.895194719739899, −1.767987352711368, −0.8203113684266995, 0,
0.8203113684266995, 1.767987352711368, 2.895194719739899, 3.284443241327431, 4.318982230269526, 4.637228090350433, 5.241342675399931, 6.044073908800863, 6.520373048176753, 7.369839623189211, 7.839430472800992, 8.307744087690806, 8.694870801794323, 9.650047934354370, 10.20374865864168, 10.84079068339231, 11.38514833358504, 11.75504584873869, 12.15219886196529, 12.46596255244797, 13.53563059133528, 14.14647620775913, 14.25905173338017, 15.13087674433756, 15.66273738058336
