| L(s) = 1 | − 2·5-s − 7-s − 11-s + 2·17-s + 2·19-s − 4·23-s − 25-s − 2·29-s + 6·31-s + 2·35-s + 5·37-s − 6·41-s + 5·43-s + 49-s + 9·53-s + 2·55-s − 2·59-s + 7·67-s + 11·71-s + 4·73-s + 77-s + 15·79-s − 10·83-s − 4·85-s + 12·89-s − 4·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.301·11-s + 0.485·17-s + 0.458·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.07·31-s + 0.338·35-s + 0.821·37-s − 0.937·41-s + 0.762·43-s + 1/7·49-s + 1.23·53-s + 0.269·55-s − 0.260·59-s + 0.855·67-s + 1.30·71-s + 0.468·73-s + 0.113·77-s + 1.68·79-s − 1.09·83-s − 0.433·85-s + 1.27·89-s − 0.410·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212196211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212196211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.997685205994559965713133471010, −8.018426868325578676881599365055, −7.70653582974649011268972075502, −6.73797830713651472136193017201, −5.91237661522044270287152415596, −5.01052820129027652717089088341, −4.03422828904062778172858859761, −3.36799044751201127158795412066, −2.27046415066059517749502915832, −0.70080249099287424881513705202,
0.70080249099287424881513705202, 2.27046415066059517749502915832, 3.36799044751201127158795412066, 4.03422828904062778172858859761, 5.01052820129027652717089088341, 5.91237661522044270287152415596, 6.73797830713651472136193017201, 7.70653582974649011268972075502, 8.018426868325578676881599365055, 8.997685205994559965713133471010
