| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 5-s + 2.41·6-s − 3.00·7-s − 4.39·8-s + 9-s − 2.41·10-s − 0.554·11-s − 3.82·12-s − 13-s + 7.24·14-s − 15-s + 2.96·16-s − 2.65·17-s − 2.41·18-s − 1.86·19-s + 3.82·20-s + 3.00·21-s + 1.33·22-s − 2.18·23-s + 4.39·24-s + 25-s + 2.41·26-s − 27-s − 11.4·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.447·5-s + 0.985·6-s − 1.13·7-s − 1.55·8-s + 0.333·9-s − 0.763·10-s − 0.167·11-s − 1.10·12-s − 0.277·13-s + 1.93·14-s − 0.258·15-s + 0.740·16-s − 0.643·17-s − 0.568·18-s − 0.427·19-s + 0.854·20-s + 0.654·21-s + 0.285·22-s − 0.455·23-s + 0.897·24-s + 0.200·25-s + 0.473·26-s − 0.192·27-s − 2.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 + 0.554T + 11T^{2} \) |
| 17 | \( 1 + 2.65T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.842T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 - 0.667T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 2.66T + 73T^{2} \) |
| 79 | \( 1 + 9.44T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.87404020362912198981390863563, −6.97368641632412662413036815061, −6.52232861158530313489587244510, −6.02907792951270350092543325664, −5.00328010706683966971370079727, −3.96166486210602823666294021954, −2.75350686413885960078442538028, −2.11100830059909226998098420042, −0.925256919980242390124310638740, 0,
0.925256919980242390124310638740, 2.11100830059909226998098420042, 2.75350686413885960078442538028, 3.96166486210602823666294021954, 5.00328010706683966971370079727, 6.02907792951270350092543325664, 6.52232861158530313489587244510, 6.97368641632412662413036815061, 7.87404020362912198981390863563
