| L(s) = 1 | − 2-s + 4-s − 8-s − 2·11-s − 6·13-s + 16-s + 4·17-s + 6·19-s + 2·22-s + 8·23-s + 6·26-s − 6·29-s + 2·31-s − 32-s − 4·34-s + 4·37-s − 6·38-s + 2·41-s + 4·43-s − 2·44-s − 8·46-s + 8·47-s − 6·52-s − 6·53-s + 6·58-s − 8·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.426·22-s + 1.66·23-s + 1.17·26-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.685·34-s + 0.657·37-s − 0.973·38-s + 0.312·41-s + 0.609·43-s − 0.301·44-s − 1.17·46-s + 1.16·47-s − 0.832·52-s − 0.824·53-s + 0.787·58-s − 1.04·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389955951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389955951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.61228798537453, −15.07223647505150, −14.46512943929990, −14.13090758492258, −13.26034759931367, −12.66799930414288, −12.29974447728861, −11.61398724799940, −11.10342555486706, −10.54774842853737, −9.755718652480575, −9.590498673798622, −9.041415857148397, −8.096642820266174, −7.654740926421526, −7.258447953125379, −6.706202377980854, −5.587190745984447, −5.362179084233239, −4.650734301578969, −3.638474687266922, −2.833293475084163, −2.466764470299364, −1.335194603183540, −0.5711502570876626,
0.5711502570876626, 1.335194603183540, 2.466764470299364, 2.833293475084163, 3.638474687266922, 4.650734301578969, 5.362179084233239, 5.587190745984447, 6.706202377980854, 7.258447953125379, 7.654740926421526, 8.096642820266174, 9.041415857148397, 9.590498673798622, 9.755718652480575, 10.54774842853737, 11.10342555486706, 11.61398724799940, 12.29974447728861, 12.66799930414288, 13.26034759931367, 14.13090758492258, 14.46512943929990, 15.07223647505150, 15.61228798537453
