| L(s) = 1 | + 5-s − 4·11-s + 5·13-s − 5·17-s + 8·19-s + 4·23-s − 4·25-s − 3·29-s + 4·31-s − 3·37-s − 6·41-s + 4·43-s + 12·47-s − 7·49-s + 10·53-s − 4·55-s + 8·59-s + 5·61-s + 5·65-s + 8·67-s − 16·71-s − 5·73-s − 4·79-s + 4·83-s − 5·85-s + 3·89-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 1.38·13-s − 1.21·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 0.557·29-s + 0.718·31-s − 0.493·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s − 49-s + 1.37·53-s − 0.539·55-s + 1.04·59-s + 0.640·61-s + 0.620·65-s + 0.977·67-s − 1.89·71-s − 0.585·73-s − 0.450·79-s + 0.439·83-s − 0.542·85-s + 0.317·89-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097888764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097888764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.327528260488543338494150059449, −7.40665600760531180895248331235, −6.88620940390391951466675863703, −5.81654181284141236056219539371, −5.51059111089904245667433238360, −4.59123022558172331067855779875, −3.63288021054679720742576555683, −2.83880787623644643075072352188, −1.92264804616084954154921526619, −0.794029449960750436345700953416,
0.794029449960750436345700953416, 1.92264804616084954154921526619, 2.83880787623644643075072352188, 3.63288021054679720742576555683, 4.59123022558172331067855779875, 5.51059111089904245667433238360, 5.81654181284141236056219539371, 6.88620940390391951466675863703, 7.40665600760531180895248331235, 8.327528260488543338494150059449
