| L(s) = 1 | + 11·2-s + 89·4-s + 627·8-s − 243·9-s − 76·11-s + 4.04e3·16-s − 2.67e3·18-s − 836·22-s − 4.95e3·23-s − 3.12e3·25-s + 7.28e3·29-s + 2.44e4·32-s − 2.16e4·36-s − 8.88e3·37-s + 1.17e4·43-s − 6.76e3·44-s − 5.44e4·46-s − 3.43e4·50-s + 2.45e4·53-s + 8.01e4·58-s + 1.39e5·64-s + 6.93e4·67-s − 2.22e3·71-s − 1.52e5·72-s − 9.77e4·74-s + 8.01e4·79-s + 5.90e4·81-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.78·4-s + 3.46·8-s − 9-s − 0.189·11-s + 3.95·16-s − 1.94·18-s − 0.368·22-s − 1.95·23-s − 25-s + 1.60·29-s + 4.22·32-s − 2.78·36-s − 1.06·37-s + 0.968·43-s − 0.526·44-s − 3.79·46-s − 1.94·50-s + 1.20·53-s + 3.12·58-s + 4.26·64-s + 1.88·67-s − 0.0523·71-s − 3.46·72-s − 2.07·74-s + 1.44·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.789031632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.789031632\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 11 T + p^{5} T^{2} \) |
| 3 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 + 76 T + p^{5} T^{2} \) |
| 13 | \( 1 + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 + 4952 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7282 T + p^{5} T^{2} \) |
| 31 | \( 1 + p^{5} T^{2} \) |
| 37 | \( 1 + 8886 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 - 11748 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 - 24550 T + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + p^{5} T^{2} \) |
| 67 | \( 1 - 69364 T + p^{5} T^{2} \) |
| 71 | \( 1 + 2224 T + p^{5} T^{2} \) |
| 73 | \( 1 + p^{5} T^{2} \) |
| 79 | \( 1 - 80168 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + p^{5} T^{2} \) |
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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
−14.20848071968594010997925807528, −13.75221005431101934413526524494, −12.33749252705086513358251775155, −11.63703710394489690966756449865, −10.37877500556039558438893736887, −7.990182933657525694443128082525, −6.40342148313154426130011603500, −5.37553365407843762437980949288, −3.87684209497451042187170083156, −2.39017691979332942802368567952,
2.39017691979332942802368567952, 3.87684209497451042187170083156, 5.37553365407843762437980949288, 6.40342148313154426130011603500, 7.990182933657525694443128082525, 10.37877500556039558438893736887, 11.63703710394489690966756449865, 12.33749252705086513358251775155, 13.75221005431101934413526524494, 14.20848071968594010997925807528
