L(s) = 1 | + 5-s − 11-s + 13-s + 4·17-s − 3·19-s + 6·23-s − 4·25-s + 7·29-s + 4·31-s − 37-s − 4·41-s − 2·43-s − 7·47-s + 10·53-s − 55-s − 9·59-s − 2·61-s + 65-s − 9·67-s − 4·71-s + 9·73-s − 16·79-s + 4·83-s + 4·85-s − 2·89-s − 3·95-s + 14·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.970·17-s − 0.688·19-s + 1.25·23-s − 4/5·25-s + 1.29·29-s + 0.718·31-s − 0.164·37-s − 0.624·41-s − 0.304·43-s − 1.02·47-s + 1.37·53-s − 0.134·55-s − 1.17·59-s − 0.256·61-s + 0.124·65-s − 1.09·67-s − 0.474·71-s + 1.05·73-s − 1.80·79-s + 0.439·83-s + 0.433·85-s − 0.211·89-s − 0.307·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.544962030\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.544962030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.70337259236620, −12.17097356681721, −11.75599500660113, −11.38740278282631, −10.58517156254679, −10.46776345704467, −9.967312256770414, −9.512422979642966, −8.948021434592188, −8.492083552386096, −8.109886152524176, −7.577617067893115, −7.017392470412496, −6.544576226291633, −6.079152101766936, −5.627862382822709, −5.024377846128510, −4.676036889771961, −4.038849232042978, −3.384887977002076, −2.913225905781362, −2.445370898543400, −1.624664311157592, −1.230696069186565, −0.4306372779603013,
0.4306372779603013, 1.230696069186565, 1.624664311157592, 2.445370898543400, 2.913225905781362, 3.384887977002076, 4.038849232042978, 4.676036889771961, 5.024377846128510, 5.627862382822709, 6.079152101766936, 6.544576226291633, 7.017392470412496, 7.577617067893115, 8.109886152524176, 8.492083552386096, 8.948021434592188, 9.512422979642966, 9.967312256770414, 10.46776345704467, 10.58517156254679, 11.38740278282631, 11.75599500660113, 12.17097356681721, 12.70337259236620