| L(s) = 1 | + 2.67·2-s − 3-s + 5.13·4-s + 3.12·5-s − 2.67·6-s − 7-s + 8.36·8-s + 9-s + 8.35·10-s − 0.191·11-s − 5.13·12-s − 0.759·13-s − 2.67·14-s − 3.12·15-s + 12.0·16-s + 2.00·17-s + 2.67·18-s − 5.29·19-s + 16.0·20-s + 21-s − 0.511·22-s + 1.40·23-s − 8.36·24-s + 4.79·25-s − 2.02·26-s − 27-s − 5.13·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.56·4-s + 1.39·5-s − 1.09·6-s − 0.377·7-s + 2.95·8-s + 0.333·9-s + 2.64·10-s − 0.0577·11-s − 1.48·12-s − 0.210·13-s − 0.713·14-s − 0.807·15-s + 3.01·16-s + 0.486·17-s + 0.629·18-s − 1.21·19-s + 3.59·20-s + 0.218·21-s − 0.109·22-s + 0.292·23-s − 1.70·24-s + 0.958·25-s − 0.397·26-s − 0.192·27-s − 0.969·28-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
| \(L(1)\) |
\(\approx\) |
\(7.091652336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.091652336\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 + 0.191T + 11T^{2} \) |
| 13 | \( 1 + 0.759T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 - 7.04T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 9.00T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 + 9.38T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 + 2.21T + 61T^{2} \) |
| 67 | \( 1 + 2.15T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 9.33T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.68T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−8.245767063854941628685180262646, −7.16091363316775342062333324753, −6.53489162639459757757900520289, −6.01033229584422746298624636892, −5.56197216413936083522867950243, −4.72382694994278370655605127165, −4.14240591195074572600621987837, −2.93447806679928685086224357965, −2.37448816412707666334994685587, −1.33195317860144286791223328202,
1.33195317860144286791223328202, 2.37448816412707666334994685587, 2.93447806679928685086224357965, 4.14240591195074572600621987837, 4.72382694994278370655605127165, 5.56197216413936083522867950243, 6.01033229584422746298624636892, 6.53489162639459757757900520289, 7.16091363316775342062333324753, 8.245767063854941628685180262646
