| L(s) = 1 | + 3-s − 5-s + 9-s + 6·13-s − 15-s − 2·19-s + 6·23-s + 25-s + 27-s + 6·29-s − 6·37-s + 6·39-s − 6·41-s − 8·43-s − 45-s − 6·47-s − 7·49-s + 6·53-s − 2·57-s + 12·59-s + 12·61-s − 6·65-s − 4·67-s + 6·69-s + 12·71-s − 2·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.960·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s − 49-s + 0.824·53-s − 0.264·57-s + 1.56·59-s + 1.53·61-s − 0.744·65-s − 0.488·67-s + 0.722·69-s + 1.42·71-s − 0.234·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.405287333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.405287333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.491558126432897481987602583832, −8.025558724206857712517754972973, −6.82033201421353101559868448454, −6.61458075237655978501027495472, −5.40631980079267041264148366817, −4.63550796724991546581859348625, −3.63216606310010362585347506817, −3.22642268183474511512896931271, −1.97776141782979305959520401409, −0.909537853146744575546185673329,
0.909537853146744575546185673329, 1.97776141782979305959520401409, 3.22642268183474511512896931271, 3.63216606310010362585347506817, 4.63550796724991546581859348625, 5.40631980079267041264148366817, 6.61458075237655978501027495472, 6.82033201421353101559868448454, 8.025558724206857712517754972973, 8.491558126432897481987602583832
