| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s + 3·11-s + 4·13-s + 2·14-s − 4·16-s − 3·19-s − 2·20-s − 6·22-s − 2·23-s + 25-s − 8·26-s − 2·28-s + 3·29-s + 8·32-s + 35-s + 2·37-s + 6·38-s + 3·41-s − 6·43-s + 6·44-s + 4·46-s − 2·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s + 0.904·11-s + 1.10·13-s + 0.534·14-s − 16-s − 0.688·19-s − 0.447·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s − 1.56·26-s − 0.377·28-s + 0.557·29-s + 1.41·32-s + 0.169·35-s + 0.328·37-s + 0.973·38-s + 0.468·41-s − 0.914·43-s + 0.904·44-s + 0.589·46-s − 0.291·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092377934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092377934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.92659695420902, −13.19296498203900, −13.00829927128508, −12.10319643414303, −11.78694267236151, −11.22646255160257, −10.82901075297727, −10.26334549370189, −9.870928984026956, −9.280589366174138, −8.756254001445967, −8.521186872876657, −7.967754597485101, −7.423646379765148, −6.789450866901667, −6.410024081477496, −5.961719468300096, −5.016964379593549, −4.368223705559436, −3.817461589150763, −3.327035152252224, −2.370854436463024, −1.793464110319654, −1.005695536643629, −0.5213876744977182,
0.5213876744977182, 1.005695536643629, 1.793464110319654, 2.370854436463024, 3.327035152252224, 3.817461589150763, 4.368223705559436, 5.016964379593549, 5.961719468300096, 6.410024081477496, 6.789450866901667, 7.423646379765148, 7.967754597485101, 8.521186872876657, 8.756254001445967, 9.280589366174138, 9.870928984026956, 10.26334549370189, 10.82901075297727, 11.22646255160257, 11.78694267236151, 12.10319643414303, 13.00829927128508, 13.19296498203900, 13.92659695420902
