| L(s) = 1 | − 0.667·2-s + 3-s − 1.55·4-s − 3.08·5-s − 0.667·6-s + 7-s + 2.37·8-s + 9-s + 2.06·10-s + 3.38·11-s − 1.55·12-s + 5.44·13-s − 0.667·14-s − 3.08·15-s + 1.52·16-s + 3.25·17-s − 0.667·18-s − 2.41·19-s + 4.80·20-s + 21-s − 2.25·22-s + 3.81·23-s + 2.37·24-s + 4.53·25-s − 3.63·26-s + 27-s − 1.55·28-s + ⋯ |
| L(s) = 1 | − 0.471·2-s + 0.577·3-s − 0.777·4-s − 1.38·5-s − 0.272·6-s + 0.377·7-s + 0.838·8-s + 0.333·9-s + 0.651·10-s + 1.01·11-s − 0.448·12-s + 1.51·13-s − 0.178·14-s − 0.797·15-s + 0.381·16-s + 0.789·17-s − 0.157·18-s − 0.555·19-s + 1.07·20-s + 0.218·21-s − 0.481·22-s + 0.795·23-s + 0.484·24-s + 0.906·25-s − 0.713·26-s + 0.192·27-s − 0.293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389605445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389605445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 0.667T + 2T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 0.602T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 0.152T + 89T^{2} \) |
| 97 | \( 1 - 4.77T + 97T^{2} \) |
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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
−8.495783099302584483959452921889, −7.933641621317417946290736312454, −7.31702329322692548808494406904, −6.43289097463874257607873876301, −5.31818371678587869618768474205, −4.37363532267366080627943257312, −3.79823010112380348057501034343, −3.35158494595164469009665762915, −1.63418989155583273688982612182, −0.77367846954041953356583027123,
0.77367846954041953356583027123, 1.63418989155583273688982612182, 3.35158494595164469009665762915, 3.79823010112380348057501034343, 4.37363532267366080627943257312, 5.31818371678587869618768474205, 6.43289097463874257607873876301, 7.31702329322692548808494406904, 7.933641621317417946290736312454, 8.495783099302584483959452921889
