| L(s) = 1 | + 2·2-s + 4·4-s − 7·5-s − 7·7-s + 8·8-s − 14·10-s + 28·11-s + 30·13-s − 14·14-s + 16·16-s − 47·17-s + 164·19-s − 28·20-s + 56·22-s + 94·23-s − 76·25-s + 60·26-s − 28·28-s + 200·29-s + 162·31-s + 32·32-s − 94·34-s + 49·35-s + 137·37-s + 328·38-s − 56·40-s − 141·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.626·5-s − 0.377·7-s + 0.353·8-s − 0.442·10-s + 0.767·11-s + 0.640·13-s − 0.267·14-s + 1/4·16-s − 0.670·17-s + 1.98·19-s − 0.313·20-s + 0.542·22-s + 0.852·23-s − 0.607·25-s + 0.452·26-s − 0.188·28-s + 1.28·29-s + 0.938·31-s + 0.176·32-s − 0.474·34-s + 0.236·35-s + 0.608·37-s + 1.40·38-s − 0.221·40-s − 0.537·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.816438189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816438189\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 47 T + p^{3} T^{2} \) |
| 19 | \( 1 - 164 T + p^{3} T^{2} \) |
| 23 | \( 1 - 94 T + p^{3} T^{2} \) |
| 29 | \( 1 - 200 T + p^{3} T^{2} \) |
| 31 | \( 1 - 162 T + p^{3} T^{2} \) |
| 37 | \( 1 - 137 T + p^{3} T^{2} \) |
| 41 | \( 1 + 141 T + p^{3} T^{2} \) |
| 43 | \( 1 - 293 T + p^{3} T^{2} \) |
| 47 | \( 1 + 471 T + p^{3} T^{2} \) |
| 53 | \( 1 - 306 T + p^{3} T^{2} \) |
| 59 | \( 1 + 331 T + p^{3} T^{2} \) |
| 61 | \( 1 + 204 T + p^{3} T^{2} \) |
| 67 | \( 1 - 928 T + p^{3} T^{2} \) |
| 71 | \( 1 + 740 T + p^{3} T^{2} \) |
| 73 | \( 1 - 706 T + p^{3} T^{2} \) |
| 79 | \( 1 + 195 T + p^{3} T^{2} \) |
| 83 | \( 1 + 485 T + p^{3} T^{2} \) |
| 89 | \( 1 + 114 T + p^{3} T^{2} \) |
| 97 | \( 1 + 344 T + p^{3} T^{2} \) |
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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
−11.36304669537755921528637115011, −10.10030993149616483306167778888, −9.115826138153707444299912862177, −8.007760839057403082994839656681, −6.98644559898270685023761211977, −6.15731955579971055096070379316, −4.91793754288985272657640630412, −3.86255189905310726948685509192, −2.92663128604992253374179337419, −1.07139603598765797827990248657,
1.07139603598765797827990248657, 2.92663128604992253374179337419, 3.86255189905310726948685509192, 4.91793754288985272657640630412, 6.15731955579971055096070379316, 6.98644559898270685023761211977, 8.007760839057403082994839656681, 9.115826138153707444299912862177, 10.10030993149616483306167778888, 11.36304669537755921528637115011
