L(s) = 1 | − 3-s − 3·5-s − 7-s − 2·9-s − 3·11-s − 2·13-s + 3·15-s + 2·17-s + 19-s + 21-s + 4·23-s + 4·25-s + 5·27-s − 2·29-s − 10·31-s + 3·33-s + 3·35-s + 37-s + 2·39-s + 5·41-s + 9·43-s + 6·45-s + 47-s + 49-s − 2·51-s − 53-s + 9·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.485·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.962·27-s − 0.371·29-s − 1.79·31-s + 0.522·33-s + 0.507·35-s + 0.164·37-s + 0.320·39-s + 0.780·41-s + 1.37·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.280·51-s − 0.137·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5253777014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5253777014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.68210818371677, −13.13267510907111, −12.64143053493839, −12.16836579910634, −11.95177291001027, −11.18933701770289, −10.83287236432298, −10.69988041080656, −9.763163173831213, −9.265047991666486, −8.837024726296776, −7.974568176282039, −7.862922042998365, −7.191194453886076, −6.884661828592859, −5.972609822312327, −5.518017757218439, −5.161168647164755, −4.418786611394743, −3.911008084057703, −3.184263783586308, −2.849109349942583, −2.050285177248228, −0.9020584898205815, −0.3004301763428371,
0.3004301763428371, 0.9020584898205815, 2.050285177248228, 2.849109349942583, 3.184263783586308, 3.911008084057703, 4.418786611394743, 5.161168647164755, 5.518017757218439, 5.972609822312327, 6.884661828592859, 7.191194453886076, 7.862922042998365, 7.974568176282039, 8.837024726296776, 9.265047991666486, 9.763163173831213, 10.69988041080656, 10.83287236432298, 11.18933701770289, 11.95177291001027, 12.16836579910634, 12.64143053493839, 13.13267510907111, 13.68210818371677