| L(s) = 1 | − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2523313857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2523313857\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| good | 2 | \( 1 + 1.80T + T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−15.30626946792415563634729611589, −13.98183065633101690550037886908, −12.42351194541329992426458232503, −11.05204135964904542931077288526, −10.37263061263202496602275828991, −9.249596659145116242975209696115, −8.392664296558968410521780245040, −6.76877928720481535930982673330, −5.76082408647750230290643880271, −2.19448912992332732423794638884,
2.19448912992332732423794638884, 5.76082408647750230290643880271, 6.76877928720481535930982673330, 8.392664296558968410521780245040, 9.249596659145116242975209696115, 10.37263061263202496602275828991, 11.05204135964904542931077288526, 12.42351194541329992426458232503, 13.98183065633101690550037886908, 15.30626946792415563634729611589
