L(s) = 1 | + 1.40·2-s + 3-s − 0.0154·4-s − 2.50·5-s + 1.40·6-s + 7-s − 2.83·8-s + 9-s − 3.52·10-s + 1.33·11-s − 0.0154·12-s + 0.606·13-s + 1.40·14-s − 2.50·15-s − 3.96·16-s + 4.81·17-s + 1.40·18-s − 4.64·19-s + 0.0385·20-s + 21-s + 1.88·22-s + 0.157·23-s − 2.83·24-s + 1.27·25-s + 0.854·26-s + 27-s − 0.0154·28-s + ⋯ |
L(s) = 1 | + 0.996·2-s + 0.577·3-s − 0.00770·4-s − 1.12·5-s + 0.575·6-s + 0.377·7-s − 1.00·8-s + 0.333·9-s − 1.11·10-s + 0.402·11-s − 0.00444·12-s + 0.168·13-s + 0.376·14-s − 0.646·15-s − 0.992·16-s + 1.16·17-s + 0.332·18-s − 1.06·19-s + 0.00862·20-s + 0.218·21-s + 0.401·22-s + 0.0329·23-s − 0.579·24-s + 0.254·25-s + 0.167·26-s + 0.192·27-s − 0.00291·28-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
\(L(1)\) |
\(\approx\) |
\(2.865250895\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.865250895\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.40T + 2T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 0.606T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 - 0.157T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 + 0.698T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 + 2.73T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.256898037719592697736873112241, −7.955640266179140709981779223456, −6.85227994881659311519438283250, −6.24094323680848354652293014500, −5.16126075290531142607315406429, −4.54519331809108308065765786532, −3.82073128468475436952919471086, −3.34881669893455129481925172860, −2.32629776415714281919626703656, −0.820366289973884015328363214326,
0.820366289973884015328363214326, 2.32629776415714281919626703656, 3.34881669893455129481925172860, 3.82073128468475436952919471086, 4.54519331809108308065765786532, 5.16126075290531142607315406429, 6.24094323680848354652293014500, 6.85227994881659311519438283250, 7.955640266179140709981779223456, 8.256898037719592697736873112241