| L(s) = 1 | + 5-s + 5·11-s − 4·17-s + 8·19-s − 4·23-s − 4·25-s + 5·29-s + 3·31-s − 4·37-s − 2·43-s + 6·47-s + 9·53-s + 5·55-s + 11·59-s + 6·61-s + 2·67-s + 2·71-s − 10·73-s − 3·79-s + 7·83-s − 4·85-s − 6·89-s + 8·95-s − 7·97-s + 10·101-s + 8·103-s + 3·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.50·11-s − 0.970·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s + 0.928·29-s + 0.538·31-s − 0.657·37-s − 0.304·43-s + 0.875·47-s + 1.23·53-s + 0.674·55-s + 1.43·59-s + 0.768·61-s + 0.244·67-s + 0.237·71-s − 1.17·73-s − 0.337·79-s + 0.768·83-s − 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.710·97-s + 0.995·101-s + 0.788·103-s + 0.290·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.554272525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554272525\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.954392288710487382545589267047, −7.08125000287789290655995260965, −6.61035124709108991851499420279, −5.85793836559137335220775065990, −5.20673819297653695881647663056, −4.22760209839302627981278703370, −3.69709615304115060651023884106, −2.66410284983016879283902355927, −1.75703840263245619039586812108, −0.852119372480057124825289539530,
0.852119372480057124825289539530, 1.75703840263245619039586812108, 2.66410284983016879283902355927, 3.69709615304115060651023884106, 4.22760209839302627981278703370, 5.20673819297653695881647663056, 5.85793836559137335220775065990, 6.61035124709108991851499420279, 7.08125000287789290655995260965, 7.954392288710487382545589267047
