| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s + 2·12-s − 13-s − 4·16-s + 7·17-s − 2·18-s − 6·19-s + 9·23-s + 2·26-s + 27-s + 4·29-s + 4·31-s + 8·32-s − 14·34-s + 2·36-s + 10·37-s + 12·38-s − 39-s + 7·41-s − 2·43-s − 18·46-s − 4·48-s + 7·51-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s + 1.87·23-s + 0.392·26-s + 0.192·27-s + 0.742·29-s + 0.718·31-s + 1.41·32-s − 2.40·34-s + 1/3·36-s + 1.64·37-s + 1.94·38-s − 0.160·39-s + 1.09·41-s − 0.304·43-s − 2.65·46-s − 0.577·48-s + 0.980·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.919348390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919348390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.62079103705115, −14.26859853229138, −13.46418808510030, −13.07071754804701, −12.50293275128014, −11.91820235335809, −11.26695646315809, −10.70383482950782, −10.31229949330419, −9.777408327398924, −9.311746147270228, −8.837513261997793, −8.296303549512404, −7.831940513830606, −7.440096549361399, −6.739305263480035, −6.293327305710980, −5.387651931594157, −4.709741608771494, −4.146117492094700, −3.266266483148334, −2.620105782665007, −2.074770126541248, −1.010766827021445, −0.7751171910159157,
0.7751171910159157, 1.010766827021445, 2.074770126541248, 2.620105782665007, 3.266266483148334, 4.146117492094700, 4.709741608771494, 5.387651931594157, 6.293327305710980, 6.739305263480035, 7.440096549361399, 7.831940513830606, 8.296303549512404, 8.837513261997793, 9.311746147270228, 9.777408327398924, 10.31229949330419, 10.70383482950782, 11.26695646315809, 11.91820235335809, 12.50293275128014, 13.07071754804701, 13.46418808510030, 14.26859853229138, 14.62079103705115
