| L(s) = 1 | − 2·4-s − 7-s + 6·11-s + 4·13-s + 4·16-s − 3·17-s + 2·19-s + 6·23-s + 2·28-s − 6·29-s − 4·31-s + 7·37-s − 3·41-s + 43-s − 12·44-s − 9·47-s + 49-s − 8·52-s + 6·53-s + 9·59-s − 10·61-s − 8·64-s + 4·67-s + 6·68-s − 2·73-s − 4·76-s − 6·77-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 1.80·11-s + 1.10·13-s + 16-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 1.15·37-s − 0.468·41-s + 0.152·43-s − 1.80·44-s − 1.31·47-s + 1/7·49-s − 1.10·52-s + 0.824·53-s + 1.17·59-s − 1.28·61-s − 64-s + 0.488·67-s + 0.727·68-s − 0.234·73-s − 0.458·76-s − 0.683·77-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
| \(L(1)\) |
\(\approx\) |
\(1.670804326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670804326\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.602732393174404237866148028168, −7.58063200157021983087517906478, −6.75859152768769293203203244131, −6.14714357596105537691088287534, −5.36312439978648140825956137198, −4.42308775811997339878637655365, −3.80054544350962618750125520790, −3.21934784703981593410979330620, −1.66436292345690568857030330571, −0.77383862239072366344800085983,
0.77383862239072366344800085983, 1.66436292345690568857030330571, 3.21934784703981593410979330620, 3.80054544350962618750125520790, 4.42308775811997339878637655365, 5.36312439978648140825956137198, 6.14714357596105537691088287534, 6.75859152768769293203203244131, 7.58063200157021983087517906478, 8.602732393174404237866148028168
