| L(s) = 1 | + 16·5-s + 12·7-s − 64·11-s + 58·13-s + 32·17-s + 136·19-s + 128·23-s + 131·25-s − 144·29-s − 20·31-s + 192·35-s − 18·37-s − 288·41-s + 200·43-s − 384·47-s − 199·49-s + 496·53-s − 1.02e3·55-s + 128·59-s − 458·61-s + 928·65-s + 496·67-s − 512·71-s − 602·73-s − 768·77-s − 1.10e3·79-s − 704·83-s + ⋯ |
| L(s) = 1 | + 1.43·5-s + 0.647·7-s − 1.75·11-s + 1.23·13-s + 0.456·17-s + 1.64·19-s + 1.16·23-s + 1.04·25-s − 0.922·29-s − 0.115·31-s + 0.927·35-s − 0.0799·37-s − 1.09·41-s + 0.709·43-s − 1.19·47-s − 0.580·49-s + 1.28·53-s − 2.51·55-s + 0.282·59-s − 0.961·61-s + 1.77·65-s + 0.904·67-s − 0.855·71-s − 0.965·73-s − 1.13·77-s − 1.57·79-s − 0.931·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.185564267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185564267\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 16 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 32 T + p^{3} T^{2} \) |
| 19 | \( 1 - 136 T + p^{3} T^{2} \) |
| 23 | \( 1 - 128 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 18 T + p^{3} T^{2} \) |
| 41 | \( 1 + 288 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 + 384 T + p^{3} T^{2} \) |
| 53 | \( 1 - 496 T + p^{3} T^{2} \) |
| 59 | \( 1 - 128 T + p^{3} T^{2} \) |
| 61 | \( 1 + 458 T + p^{3} T^{2} \) |
| 67 | \( 1 - 496 T + p^{3} T^{2} \) |
| 71 | \( 1 + 512 T + p^{3} T^{2} \) |
| 73 | \( 1 + 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1108 T + p^{3} T^{2} \) |
| 83 | \( 1 + 704 T + p^{3} T^{2} \) |
| 89 | \( 1 + 960 T + p^{3} T^{2} \) |
| 97 | \( 1 - 206 T + p^{3} T^{2} \) |
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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
−13.04196104598443242166423960320, −11.43857449836526423769531808224, −10.52801083806468471276326692906, −9.653716853892745682149680744605, −8.483433385867203710990513384255, −7.32489179449346648458901559504, −5.74213692315212629316764765185, −5.16315587640477433054201822521, −2.99867682808167231410755433430, −1.47259866721317812782581378688,
1.47259866721317812782581378688, 2.99867682808167231410755433430, 5.16315587640477433054201822521, 5.74213692315212629316764765185, 7.32489179449346648458901559504, 8.483433385867203710990513384255, 9.653716853892745682149680744605, 10.52801083806468471276326692906, 11.43857449836526423769531808224, 13.04196104598443242166423960320
