| L(s) = 1 | − 1.24·2-s + 0.445·3-s − 0.445·4-s + 0.554·5-s − 0.554·6-s + 1.35·7-s + 3.04·8-s − 2.80·9-s − 0.692·10-s − 2.15·11-s − 0.198·12-s + 1.58·13-s − 1.69·14-s + 0.246·15-s − 2.91·16-s + 2.49·17-s + 3.49·18-s − 5.80·19-s − 0.246·20-s + 0.603·21-s + 2.69·22-s + 5.09·23-s + 1.35·24-s − 4.69·25-s − 1.97·26-s − 2.58·27-s − 0.603·28-s + ⋯ |
| L(s) = 1 | − 0.881·2-s + 0.256·3-s − 0.222·4-s + 0.248·5-s − 0.226·6-s + 0.512·7-s + 1.07·8-s − 0.933·9-s − 0.218·10-s − 0.650·11-s − 0.0571·12-s + 0.438·13-s − 0.452·14-s + 0.0637·15-s − 0.727·16-s + 0.604·17-s + 0.823·18-s − 1.33·19-s − 0.0552·20-s + 0.131·21-s + 0.573·22-s + 1.06·23-s + 0.276·24-s − 0.938·25-s − 0.386·26-s − 0.496·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 - 0.445T + 3T^{2} \) |
| 5 | \( 1 - 0.554T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7.62T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 - 0.917T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.800938604878146511455396915692, −8.140244197038614680640686366595, −7.75830860223088875562133578699, −6.56851472577406863351431825324, −5.58402338738749320916043821979, −4.83687610768424297391441365176, −3.77653547329191098204826637546, −2.57744811110109640540126886464, −1.49080644779524810862631328236, 0,
1.49080644779524810862631328236, 2.57744811110109640540126886464, 3.77653547329191098204826637546, 4.83687610768424297391441365176, 5.58402338738749320916043821979, 6.56851472577406863351431825324, 7.75830860223088875562133578699, 8.140244197038614680640686366595, 8.800938604878146511455396915692
