| L(s) = 1 | + 3-s + 7-s + 9-s − 2·13-s − 6·17-s − 4·19-s + 21-s + 4·23-s + 27-s − 6·29-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 6·51-s + 14·53-s − 4·57-s − 4·59-s + 2·61-s + 63-s − 12·67-s + 4·69-s + 12·71-s − 10·73-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.92·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.481·69-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434130917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434130917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.03546579775222, −14.59335528567435, −13.96494549976840, −13.34989147073508, −13.00462269852494, −12.52298066972466, −11.77162039909229, −11.25576799952117, −10.70402679045614, −10.31912984761371, −9.417236844753850, −9.023188654068974, −8.703031067097705, −7.880096787078670, −7.416174978099193, −6.912509740538590, −6.206311212064544, −5.568973958910915, −4.791258734257493, −4.235807175303473, −3.818673490257556, −2.619820288909873, −2.453519454815927, −1.586467878564738, −0.5549893645345528,
0.5549893645345528, 1.586467878564738, 2.453519454815927, 2.619820288909873, 3.818673490257556, 4.235807175303473, 4.791258734257493, 5.568973958910915, 6.206311212064544, 6.912509740538590, 7.416174978099193, 7.880096787078670, 8.703031067097705, 9.023188654068974, 9.417236844753850, 10.31912984761371, 10.70402679045614, 11.25576799952117, 11.77162039909229, 12.52298066972466, 13.00462269852494, 13.34989147073508, 13.96494549976840, 14.59335528567435, 15.03546579775222
