| L(s) = 1 | − 5-s − 4·7-s − 11-s − 13-s + 2·17-s + 4·19-s + 25-s − 6·29-s − 8·31-s + 4·35-s − 2·37-s − 2·41-s + 8·43-s + 8·47-s + 9·49-s + 10·53-s + 55-s + 12·59-s + 14·61-s + 65-s + 4·67-s − 8·71-s − 2·73-s + 4·77-s − 2·85-s − 10·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.312·41-s + 1.21·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.134·55-s + 1.56·59-s + 1.79·61-s + 0.124·65-s + 0.488·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.216·85-s − 1.05·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098657811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098657811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.49694883173650, −13.25845596339574, −12.75330386665145, −12.26053461002465, −11.95729499957634, −11.19323358198146, −10.87337805580462, −10.13084164951707, −9.783896398546944, −9.399992992532337, −8.783752502650389, −8.338034822586514, −7.460233148989261, −7.204415669040377, −6.910950205687549, −5.984412522476705, −5.544586832730544, −5.274170262316585, −4.191847706768801, −3.789650517697526, −3.341810398980346, −2.684324140356461, −2.126674838837185, −1.061842991253747, −0.3702337074846911,
0.3702337074846911, 1.061842991253747, 2.126674838837185, 2.684324140356461, 3.341810398980346, 3.789650517697526, 4.191847706768801, 5.274170262316585, 5.544586832730544, 5.984412522476705, 6.910950205687549, 7.204415669040377, 7.460233148989261, 8.338034822586514, 8.783752502650389, 9.399992992532337, 9.783896398546944, 10.13084164951707, 10.87337805580462, 11.19323358198146, 11.95729499957634, 12.26053461002465, 12.75330386665145, 13.25845596339574, 13.49694883173650
