| L(s) = 1 | − 2·2-s + 2·4-s + 7-s + 4·11-s + 2·13-s − 2·14-s − 4·16-s + 3·17-s − 8·19-s − 8·22-s − 6·23-s − 4·26-s + 2·28-s + 4·29-s + 6·31-s + 8·32-s − 6·34-s + 3·37-s + 16·38-s − 41-s − 11·43-s + 8·44-s + 12·46-s + 9·47-s + 49-s + 4·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.534·14-s − 16-s + 0.727·17-s − 1.83·19-s − 1.70·22-s − 1.25·23-s − 0.784·26-s + 0.377·28-s + 0.742·29-s + 1.07·31-s + 1.41·32-s − 1.02·34-s + 0.493·37-s + 2.59·38-s − 0.156·41-s − 1.67·43-s + 1.20·44-s + 1.76·46-s + 1.31·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012902154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012902154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.407278654361839594875042079713, −7.969849777084856880127752292939, −6.81502226261946169031511424942, −6.54633235757052815762147036572, −5.55980524920676678159441673636, −4.34985966101146994294648839607, −3.89790184589722926265726875919, −2.45382424012958058933057868357, −1.61818169107176743751082127937, −0.73231116309430509246980233589,
0.73231116309430509246980233589, 1.61818169107176743751082127937, 2.45382424012958058933057868357, 3.89790184589722926265726875919, 4.34985966101146994294648839607, 5.55980524920676678159441673636, 6.54633235757052815762147036572, 6.81502226261946169031511424942, 7.969849777084856880127752292939, 8.407278654361839594875042079713
