| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 2·15-s + 17-s + 6·19-s + 2·21-s + 6·23-s − 25-s − 27-s + 2·29-s − 4·31-s + 4·35-s − 10·37-s + 6·41-s − 10·43-s − 2·45-s + 12·47-s − 3·49-s − 51-s − 4·53-s − 6·57-s + 10·59-s − 10·61-s − 2·63-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.516·15-s + 0.242·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.937·41-s − 1.52·43-s − 0.298·45-s + 1.75·47-s − 3/7·49-s − 0.140·51-s − 0.549·53-s − 0.794·57-s + 1.30·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005287351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005287351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.59321817676529, −13.31737806848482, −12.59817687551550, −12.32286497667302, −11.77394614331246, −11.42285410753719, −10.91280532138343, −10.30908881636379, −9.924229642101808, −9.299246982310009, −8.851214729804420, −8.279911474101619, −7.519249893109726, −7.197142364084655, −6.895608439381806, −5.996295134900744, −5.700034418194694, −4.948637645031018, −4.595114695384807, −3.693559092915714, −3.387929289065315, −2.866007814532416, −1.854947508214207, −1.063816349846414, −0.3811650409238801,
0.3811650409238801, 1.063816349846414, 1.854947508214207, 2.866007814532416, 3.387929289065315, 3.693559092915714, 4.595114695384807, 4.948637645031018, 5.700034418194694, 5.996295134900744, 6.895608439381806, 7.197142364084655, 7.519249893109726, 8.279911474101619, 8.851214729804420, 9.299246982310009, 9.924229642101808, 10.30908881636379, 10.91280532138343, 11.42285410753719, 11.77394614331246, 12.32286497667302, 12.59817687551550, 13.31737806848482, 13.59321817676529
