| L(s) = 1 | + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s + 6·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s − 7·49-s + 2·53-s + 4·59-s + 2·61-s − 4·67-s − 8·71-s − 10·73-s − 8·79-s + 4·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s + 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.462195988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.462195988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
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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
−16.19058144110895, −15.67900503872922, −14.82995334932194, −14.42434393176136, −13.97508063149394, −13.46109426933564, −12.60395817036435, −12.00279684373947, −11.81085764772240, −11.14302197596631, −10.16243853308940, −9.907977351810302, −9.327363300660975, −8.587109362109026, −7.949763960722478, −7.435776479563492, −6.633300323571947, −6.106701629919368, −5.508754746463744, −4.477023250831347, −4.198213512086672, −3.204322549905620, −2.551556965452816, −1.522730366924427, −0.7336646007333192,
0.7336646007333192, 1.522730366924427, 2.551556965452816, 3.204322549905620, 4.198213512086672, 4.477023250831347, 5.508754746463744, 6.106701629919368, 6.633300323571947, 7.435776479563492, 7.949763960722478, 8.587109362109026, 9.327363300660975, 9.907977351810302, 10.16243853308940, 11.14302197596631, 11.81085764772240, 12.00279684373947, 12.60395817036435, 13.46109426933564, 13.97508063149394, 14.42434393176136, 14.82995334932194, 15.67900503872922, 16.19058144110895
