| L(s) = 1 | − 2-s + 3.04·3-s + 4-s + 1.48·5-s − 3.04·6-s + 0.344·7-s − 8-s + 6.29·9-s − 1.48·10-s − 0.905·11-s + 3.04·12-s − 2.24·13-s − 0.344·14-s + 4.53·15-s + 16-s + 2.58·17-s − 6.29·18-s − 0.688·19-s + 1.48·20-s + 1.04·21-s + 0.905·22-s + 1.46·23-s − 3.04·24-s − 2.78·25-s + 2.24·26-s + 10.0·27-s + 0.344·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s + 0.665·5-s − 1.24·6-s + 0.130·7-s − 0.353·8-s + 2.09·9-s − 0.470·10-s − 0.273·11-s + 0.880·12-s − 0.623·13-s − 0.0919·14-s + 1.17·15-s + 0.250·16-s + 0.627·17-s − 1.48·18-s − 0.157·19-s + 0.332·20-s + 0.229·21-s + 0.193·22-s + 0.304·23-s − 0.622·24-s − 0.557·25-s + 0.441·26-s + 1.93·27-s + 0.0650·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067539681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067539681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 - 0.344T + 7T^{2} \) |
| 11 | \( 1 + 0.905T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 0.688T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 6.59T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−10.27294859026301331102160398984, −9.772291867283740368337284756728, −9.096558459979297704909355160644, −8.200609362081373821187882845746, −7.62246017138901322808141009128, −6.63178167518341311723693321948, −5.18577238747919645306811178745, −3.68149435561549818431328879875, −2.61259424255856168354097745297, −1.71212161812236731954651064958,
1.71212161812236731954651064958, 2.61259424255856168354097745297, 3.68149435561549818431328879875, 5.18577238747919645306811178745, 6.63178167518341311723693321948, 7.62246017138901322808141009128, 8.200609362081373821187882845746, 9.096558459979297704909355160644, 9.772291867283740368337284756728, 10.27294859026301331102160398984
