| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s − 15-s + 16-s − 6·17-s + 18-s + 2·19-s − 20-s + 2·22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 30-s + 32-s + 2·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.182·30-s + 0.176·32-s + 0.348·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.183830785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.183830785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.65604878993023, −11.96948016366336, −11.55642257783876, −11.36001638982399, −10.80051203570742, −10.30120999795840, −9.719296101289793, −9.253385635080323, −8.900795388035265, −8.298485688352970, −7.901883051213080, −7.448433268180112, −6.903017510315264, −6.405761205031772, −6.151017706782313, −5.455838148691379, −4.704831374188386, −4.474092802689099, −4.047047595938181, −3.381877367328443, −3.028891562865721, −2.452516480530091, −1.767135129127413, −1.335187555233984, −0.4285616437611273,
0.4285616437611273, 1.335187555233984, 1.767135129127413, 2.452516480530091, 3.028891562865721, 3.381877367328443, 4.047047595938181, 4.474092802689099, 4.704831374188386, 5.455838148691379, 6.151017706782313, 6.405761205031772, 6.903017510315264, 7.448433268180112, 7.901883051213080, 8.298485688352970, 8.900795388035265, 9.253385635080323, 9.719296101289793, 10.30120999795840, 10.80051203570742, 11.36001638982399, 11.55642257783876, 11.96948016366336, 12.65604878993023
