| L(s) = 1 | + 1.29·2-s + 3·3-s − 6.31·4-s + 7.05·5-s + 3.89·6-s − 33.4·7-s − 18.5·8-s + 9·9-s + 9.15·10-s − 27.5·11-s − 18.9·12-s − 55.6·13-s − 43.4·14-s + 21.1·15-s + 26.3·16-s + 108.·17-s + 11.6·18-s − 141.·19-s − 44.5·20-s − 100.·21-s − 35.8·22-s + 142.·23-s − 55.7·24-s − 75.2·25-s − 72.2·26-s + 27·27-s + 211.·28-s + ⋯ |
| L(s) = 1 | + 0.458·2-s + 0.577·3-s − 0.789·4-s + 0.631·5-s + 0.264·6-s − 1.80·7-s − 0.821·8-s + 0.333·9-s + 0.289·10-s − 0.756·11-s − 0.455·12-s − 1.18·13-s − 0.828·14-s + 0.364·15-s + 0.412·16-s + 1.54·17-s + 0.152·18-s − 1.70·19-s − 0.498·20-s − 1.04·21-s − 0.347·22-s + 1.29·23-s − 0.474·24-s − 0.601·25-s − 0.545·26-s + 0.192·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 - 1.29T + 8T^{2} \) |
| 5 | \( 1 - 7.05T + 125T^{2} \) |
| 7 | \( 1 + 33.4T + 343T^{2} \) |
| 11 | \( 1 + 27.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 97.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 380.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 15.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 172.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 616.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 210.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 543.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 350.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 625.T + 9.12e5T^{2} \) |
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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
−12.31201052886129989050763436039, −10.27882062286846095309248542703, −9.712266695242400467227604220773, −8.968091682787389481530675152872, −7.55687599673244993489344278411, −6.23391131525252373794030154958, −5.19269740561505549037638280512, −3.67390495373758240107983465066, −2.65515941951464657731438894273, 0,
2.65515941951464657731438894273, 3.67390495373758240107983465066, 5.19269740561505549037638280512, 6.23391131525252373794030154958, 7.55687599673244993489344278411, 8.968091682787389481530675152872, 9.712266695242400467227604220773, 10.27882062286846095309248542703, 12.31201052886129989050763436039
