| L(s) = 1 | − 5-s − 4·7-s + 4·13-s − 4·19-s − 6·23-s + 25-s − 6·29-s − 8·31-s + 4·35-s + 2·37-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·53-s − 12·59-s − 2·61-s − 4·65-s + 10·67-s − 12·71-s + 16·73-s + 8·79-s − 6·89-s − 16·91-s + 4·95-s + 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.496·65-s + 1.22·67-s − 1.42·71-s + 1.87·73-s + 0.900·79-s − 0.635·89-s − 1.67·91-s + 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6414894341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6414894341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.83779814703676, −13.34279230106179, −12.81552375093620, −12.55268745752804, −12.09001931243545, −11.33506513580107, −10.80587488488927, −10.63699196615051, −9.814517853778296, −9.370175908890980, −8.974620077787424, −8.450479317740666, −7.685394615646233, −7.410468111033469, −6.659298498155139, −6.086060775561553, −5.947582043121594, −5.188068305376285, −4.172941082692343, −3.855156100296041, −3.541621221620832, −2.649594639108158, −2.131679509416876, −1.181393891306063, −0.2716308894760162,
0.2716308894760162, 1.181393891306063, 2.131679509416876, 2.649594639108158, 3.541621221620832, 3.855156100296041, 4.172941082692343, 5.188068305376285, 5.947582043121594, 6.086060775561553, 6.659298498155139, 7.410468111033469, 7.685394615646233, 8.450479317740666, 8.974620077787424, 9.370175908890980, 9.814517853778296, 10.63699196615051, 10.80587488488927, 11.33506513580107, 12.09001931243545, 12.55268745752804, 12.81552375093620, 13.34279230106179, 13.83779814703676
