| L(s) = 1 | + 5-s + 3·7-s − 3·9-s + 5·11-s − 7·17-s − 4·23-s − 4·25-s + 3·35-s + 43-s − 3·45-s + 13·47-s + 2·49-s + 5·55-s − 15·61-s − 9·63-s − 11·73-s + 15·77-s + 9·81-s + 16·83-s − 7·85-s − 15·99-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 9-s + 1.50·11-s − 1.69·17-s − 0.834·23-s − 4/5·25-s + 0.507·35-s + 0.152·43-s − 0.447·45-s + 1.89·47-s + 2/7·49-s + 0.674·55-s − 1.92·61-s − 1.13·63-s − 1.28·73-s + 1.70·77-s + 81-s + 1.75·83-s − 0.759·85-s − 1.50·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.56839485069380, −15.17230349347647, −14.52317172748590, −14.15130228690412, −13.71436392907902, −13.28391777861603, −12.19300053145329, −11.96051148328597, −11.38069768394070, −10.93570174149197, −10.41347264861475, −9.443794368809015, −9.092956546288065, −8.629390887500896, −8.011352220756035, −7.362040876502239, −6.572590303737043, −6.075856607537235, −5.593405741315282, −4.674078741636069, −4.258099067342318, −3.551055080698526, −2.470107466540217, −1.978925811172291, −1.224548029807122, 0,
1.224548029807122, 1.978925811172291, 2.470107466540217, 3.551055080698526, 4.258099067342318, 4.674078741636069, 5.593405741315282, 6.075856607537235, 6.572590303737043, 7.362040876502239, 8.011352220756035, 8.629390887500896, 9.092956546288065, 9.443794368809015, 10.41347264861475, 10.93570174149197, 11.38069768394070, 11.96051148328597, 12.19300053145329, 13.28391777861603, 13.71436392907902, 14.15130228690412, 14.52317172748590, 15.17230349347647, 15.56839485069380
