| L(s) = 1 | − 2.80·2-s + 0.290·3-s + 5.88·4-s − 1.66·5-s − 0.815·6-s + 1.40·7-s − 10.8·8-s − 2.91·9-s + 4.66·10-s + 5.00·11-s + 1.70·12-s − 2.19·13-s − 3.95·14-s − 0.482·15-s + 18.8·16-s + 5.06·17-s + 8.18·18-s − 5.60·19-s − 9.77·20-s + 0.409·21-s − 14.0·22-s + 7.70·23-s − 3.16·24-s − 2.23·25-s + 6.16·26-s − 1.71·27-s + 8.28·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.167·3-s + 2.94·4-s − 0.743·5-s − 0.332·6-s + 0.532·7-s − 3.85·8-s − 0.971·9-s + 1.47·10-s + 1.50·11-s + 0.493·12-s − 0.608·13-s − 1.05·14-s − 0.124·15-s + 4.70·16-s + 1.22·17-s + 1.92·18-s − 1.28·19-s − 2.18·20-s + 0.0893·21-s − 2.99·22-s + 1.60·23-s − 0.646·24-s − 0.447·25-s + 1.20·26-s − 0.330·27-s + 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6943052786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6943052786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.290T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 - 1.40T + 7T^{2} \) |
| 11 | \( 1 - 5.00T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 0.908T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 - 0.206T + 43T^{2} \) |
| 47 | \( 1 - 2.86T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 8.14T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 0.357T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.657827145309776899503131132509, −7.76956433612094184349378932445, −7.47140481785759573463773320481, −6.44664198986435360493971374530, −6.01354430153721989056873697654, −4.65155146402502784742147738358, −3.38751948464198731536006395350, −2.70236363278587686707594146806, −1.57357731652485324165954164556, −0.66164270782059726239354908238,
0.66164270782059726239354908238, 1.57357731652485324165954164556, 2.70236363278587686707594146806, 3.38751948464198731536006395350, 4.65155146402502784742147738358, 6.01354430153721989056873697654, 6.44664198986435360493971374530, 7.47140481785759573463773320481, 7.76956433612094184349378932445, 8.657827145309776899503131132509
