L(s) = 1 | + 1.23·2-s + 0.357·3-s − 0.483·4-s + 1.04·5-s + 0.439·6-s + 4.82·7-s − 3.05·8-s − 2.87·9-s + 1.28·10-s − 11-s − 0.172·12-s + 13-s + 5.94·14-s + 0.372·15-s − 2.79·16-s − 2.46·17-s − 3.53·18-s − 2.68·19-s − 0.505·20-s + 1.72·21-s − 1.23·22-s − 6.84·23-s − 1.09·24-s − 3.90·25-s + 1.23·26-s − 2.09·27-s − 2.33·28-s + ⋯ |
L(s) = 1 | + 0.870·2-s + 0.206·3-s − 0.241·4-s + 0.467·5-s + 0.179·6-s + 1.82·7-s − 1.08·8-s − 0.957·9-s + 0.406·10-s − 0.301·11-s − 0.0498·12-s + 0.277·13-s + 1.58·14-s + 0.0962·15-s − 0.699·16-s − 0.597·17-s − 0.833·18-s − 0.616·19-s − 0.113·20-s + 0.376·21-s − 0.262·22-s − 1.42·23-s − 0.222·24-s − 0.781·25-s + 0.241·26-s − 0.403·27-s − 0.441·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.688293950\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688293950\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 3 | \( 1 - 0.357T + 3T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 - 0.450T + 37T^{2} \) |
| 41 | \( 1 + 0.139T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 3.74T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 - 5.87T + 61T^{2} \) |
| 67 | \( 1 - 6.01T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 - 7.06T + 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−13.61827259333092628939331305036, −12.05978409721013444605731101370, −11.43713176679695830018391218094, −10.15571583228672969549684988675, −8.624937747076016596178105140982, −8.163323367372863023764371547033, −6.17564516972940277475218634181, −5.18938021307411927282979509842, −4.18484703627831678227494085134, −2.34232815520068844557905736663,
2.34232815520068844557905736663, 4.18484703627831678227494085134, 5.18938021307411927282979509842, 6.17564516972940277475218634181, 8.163323367372863023764371547033, 8.624937747076016596178105140982, 10.15571583228672969549684988675, 11.43713176679695830018391218094, 12.05978409721013444605731101370, 13.61827259333092628939331305036