| L(s) = 1 | + 2·3-s + 5-s + 9-s − 11-s − 5·13-s + 2·15-s − 6·17-s + 4·19-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 5·31-s − 2·33-s − 4·37-s − 10·39-s + 11·43-s + 45-s − 12·51-s + 6·53-s − 55-s + 8·57-s + 3·59-s + 10·61-s − 5·65-s + 8·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.898·31-s − 0.348·33-s − 0.657·37-s − 1.60·39-s + 1.67·43-s + 0.149·45-s − 1.68·51-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + 0.390·59-s + 1.28·61-s − 0.620·65-s + 0.977·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.048947400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.048947400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.56498732365417, −15.64074797683760, −15.45505941012801, −14.66700414437776, −14.24620267803506, −13.83404837319289, −13.09112008097120, −12.78519382181522, −11.96457544536144, −11.27996261977736, −10.62048311676167, −9.958508623989727, −9.259388456087513, −9.029178696651028, −8.319207749704809, −7.554474704382097, −7.101707939523671, −6.424573057778801, −5.314897012946307, −4.986673687953623, −4.023327674028319, −3.203979020747314, −2.424192295795010, −2.160084885550523, −0.7373899276472023,
0.7373899276472023, 2.160084885550523, 2.424192295795010, 3.203979020747314, 4.023327674028319, 4.986673687953623, 5.314897012946307, 6.424573057778801, 7.101707939523671, 7.554474704382097, 8.319207749704809, 9.029178696651028, 9.259388456087513, 9.958508623989727, 10.62048311676167, 11.27996261977736, 11.96457544536144, 12.78519382181522, 13.09112008097120, 13.83404837319289, 14.24620267803506, 14.66700414437776, 15.45505941012801, 15.64074797683760, 16.56498732365417
