| L(s) = 1 | + 16·3-s + 54·5-s + 112·7-s + 13·9-s + 472·11-s − 290·13-s + 864·15-s + 738·17-s + 1.61e3·19-s + 1.79e3·21-s − 1.50e3·23-s − 209·25-s − 3.68e3·27-s + 841·29-s + 5.32e3·31-s + 7.55e3·33-s + 6.04e3·35-s − 2.65e3·37-s − 4.64e3·39-s − 4.47e3·41-s + 2.82e3·43-s + 702·45-s + 3.02e3·47-s − 4.26e3·49-s + 1.18e4·51-s + 5.57e3·53-s + 2.54e4·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + 0.965·5-s + 0.863·7-s + 0.0534·9-s + 1.17·11-s − 0.475·13-s + 0.991·15-s + 0.619·17-s + 1.02·19-s + 0.886·21-s − 0.592·23-s − 0.0668·25-s − 0.971·27-s + 0.185·29-s + 0.995·31-s + 1.20·33-s + 0.834·35-s − 0.318·37-s − 0.488·39-s − 0.415·41-s + 0.232·43-s + 0.0516·45-s + 0.199·47-s − 0.253·49-s + 0.635·51-s + 0.272·53-s + 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.028805154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.028805154\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 16 T + p^{5} T^{2} \) |
| 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 16 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 472 T + p^{5} T^{2} \) |
| 13 | \( 1 + 290 T + p^{5} T^{2} \) |
| 17 | \( 1 - 738 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1616 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1504 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5324 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2650 T + p^{5} T^{2} \) |
| 41 | \( 1 + 4470 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2824 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3028 T + p^{5} T^{2} \) |
| 53 | \( 1 - 5574 T + p^{5} T^{2} \) |
| 59 | \( 1 + 22764 T + p^{5} T^{2} \) |
| 61 | \( 1 - 654 T + p^{5} T^{2} \) |
| 67 | \( 1 - 54612 T + p^{5} T^{2} \) |
| 71 | \( 1 - 5480 T + p^{5} T^{2} \) |
| 73 | \( 1 - 49370 T + p^{5} T^{2} \) |
| 79 | \( 1 + 8020 T + p^{5} T^{2} \) |
| 83 | \( 1 + 17508 T + p^{5} T^{2} \) |
| 89 | \( 1 - 63114 T + p^{5} T^{2} \) |
| 97 | \( 1 + 5614 T + p^{5} T^{2} \) |
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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
−11.42878101763975861872154929200, −10.00516194975076944773943090057, −9.397041135856737501662605034801, −8.436317796291983633205937662307, −7.54510390232260314098395277134, −6.20286586008432040470236871643, −5.06369308461580888703003333871, −3.63752266667561819933473349813, −2.34928251894986016442599886082, −1.31716455583486471745801249357,
1.31716455583486471745801249357, 2.34928251894986016442599886082, 3.63752266667561819933473349813, 5.06369308461580888703003333871, 6.20286586008432040470236871643, 7.54510390232260314098395277134, 8.436317796291983633205937662307, 9.397041135856737501662605034801, 10.00516194975076944773943090057, 11.42878101763975861872154929200
