| L(s) = 1 | − 1.61·2-s − 0.381·3-s + 0.618·4-s + 0.618·6-s − 3·7-s + 2.23·8-s − 2.85·9-s − 3·11-s − 0.236·12-s − 4.85·13-s + 4.85·14-s − 4.85·16-s + 4.23·17-s + 4.61·18-s − 3.61·19-s + 1.14·21-s + 4.85·22-s − 1.23·23-s − 0.854·24-s + 7.85·26-s + 2.23·27-s − 1.85·28-s + 6.70·29-s + 5.09·31-s + 3.38·32-s + 1.14·33-s − 6.85·34-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.220·3-s + 0.309·4-s + 0.252·6-s − 1.13·7-s + 0.790·8-s − 0.951·9-s − 0.904·11-s − 0.0681·12-s − 1.34·13-s + 1.29·14-s − 1.21·16-s + 1.02·17-s + 1.08·18-s − 0.830·19-s + 0.250·21-s + 1.03·22-s − 0.257·23-s − 0.174·24-s + 1.54·26-s + 0.430·27-s − 0.350·28-s + 1.24·29-s + 0.914·31-s + 0.597·32-s + 0.199·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 - 0.472T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
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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
−12.77530826107366461194664135181, −11.73792435979112215104379094148, −10.26038123706751969123599255840, −9.939295121411511505824047360314, −8.647305319067773250473627408691, −7.72847099897749536043262283778, −6.42548266234641240966635633122, −4.94771465460436105819958449903, −2.81326367837647123037569197004, 0,
2.81326367837647123037569197004, 4.94771465460436105819958449903, 6.42548266234641240966635633122, 7.72847099897749536043262283778, 8.647305319067773250473627408691, 9.939295121411511505824047360314, 10.26038123706751969123599255840, 11.73792435979112215104379094148, 12.77530826107366461194664135181
