| L(s) = 1 | − 2.64·3-s − 3.64·5-s + 4.00·9-s − 11-s − 5·13-s + 9.64·15-s − 6·17-s − 0.354·19-s + 3.64·23-s + 8.29·25-s − 2.64·27-s − 4.29·29-s − 4·31-s + 2.64·33-s − 1.64·37-s + 13.2·39-s + 4.93·41-s + 4·43-s − 14.5·45-s + 13.2·47-s + 15.8·51-s − 3.64·53-s + 3.64·55-s + 0.937·57-s − 0.645·59-s − 3.70·61-s + 18.2·65-s + ⋯ |
| L(s) = 1 | − 1.52·3-s − 1.63·5-s + 1.33·9-s − 0.301·11-s − 1.38·13-s + 2.49·15-s − 1.45·17-s − 0.0812·19-s + 0.760·23-s + 1.65·25-s − 0.509·27-s − 0.796·29-s − 0.718·31-s + 0.460·33-s − 0.270·37-s + 2.11·39-s + 0.771·41-s + 0.609·43-s − 2.17·45-s + 1.93·47-s + 2.22·51-s − 0.500·53-s + 0.491·55-s + 0.124·57-s − 0.0840·59-s − 0.474·61-s + 2.26·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 0.354T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 0.645T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 3.93T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−7.33363335618312236867250231983, −6.90234181715891107754414192100, −6.01064684262374729308619919963, −5.27571238709730677473179023448, −4.53494502552171281607073735717, −4.29212569325188369180551388014, −3.19610847981346962800495340881, −2.16515675850196788961496420223, −0.67269594652445934033221809223, 0,
0.67269594652445934033221809223, 2.16515675850196788961496420223, 3.19610847981346962800495340881, 4.29212569325188369180551388014, 4.53494502552171281607073735717, 5.27571238709730677473179023448, 6.01064684262374729308619919963, 6.90234181715891107754414192100, 7.33363335618312236867250231983
