| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s + 11-s + 4·13-s − 4·15-s + 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s − 2·29-s + 10·31-s + 2·33-s − 2·35-s + 6·37-s + 8·39-s − 4·43-s − 2·45-s − 10·47-s + 49-s + 14·53-s − 2·55-s + 8·57-s + 10·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.03·15-s + 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s + 1.79·31-s + 0.348·33-s − 0.338·35-s + 0.986·37-s + 1.28·39-s − 0.609·43-s − 0.298·45-s − 1.45·47-s + 1/7·49-s + 1.92·53-s − 0.269·55-s + 1.05·57-s + 1.30·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.788522228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.788522228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.304678966850277604677440259742, −7.83083974083978329028774521583, −7.02290344723884324202509792065, −6.15691739492390898458425685818, −5.28251631768211194433493567459, −4.16842783255875351041500627817, −3.76032993844132753153824064245, −2.98813288050884403529808457607, −2.03596941501175028314745945381, −0.889259241892590930272251578396,
0.889259241892590930272251578396, 2.03596941501175028314745945381, 2.98813288050884403529808457607, 3.76032993844132753153824064245, 4.16842783255875351041500627817, 5.28251631768211194433493567459, 6.15691739492390898458425685818, 7.02290344723884324202509792065, 7.83083974083978329028774521583, 8.304678966850277604677440259742
