L(s) = 1 | − 0.0954·3-s − 0.581·5-s − 2.99·9-s − 4.59·11-s − 5.44·13-s + 0.0554·15-s − 0.983·17-s + 0.599·19-s + 5.72·23-s − 4.66·25-s + 0.571·27-s + 6.69·29-s + 1.57·31-s + 0.438·33-s − 8.89·37-s + 0.519·39-s − 41-s − 5.24·43-s + 1.73·45-s − 7.66·47-s + 0.0938·51-s − 11.3·53-s + 2.66·55-s − 0.0572·57-s + 7.53·59-s + 5.90·61-s + 3.16·65-s + ⋯ |
L(s) = 1 | − 0.0550·3-s − 0.259·5-s − 0.996·9-s − 1.38·11-s − 1.51·13-s + 0.0143·15-s − 0.238·17-s + 0.137·19-s + 1.19·23-s − 0.932·25-s + 0.110·27-s + 1.24·29-s + 0.283·31-s + 0.0762·33-s − 1.46·37-s + 0.0832·39-s − 0.156·41-s − 0.799·43-s + 0.259·45-s − 1.11·47-s + 0.0131·51-s − 1.55·53-s + 0.359·55-s − 0.00758·57-s + 0.980·59-s + 0.755·61-s + 0.392·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6123998613\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6123998613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 0.0954T + 3T^{2} \) |
| 5 | \( 1 + 0.581T + 5T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 17 | \( 1 + 0.983T + 17T^{2} \) |
| 19 | \( 1 - 0.599T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 - 1.57T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 5.90T + 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 + 8.01T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 1.44T + 97T^{2} \) |
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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
−7.945297023190132210941362505631, −7.14922111252467631052199215501, −6.57006288553514176971124892132, −5.52185229862619635051990862026, −5.09927947872366767473327366525, −4.51632642511709227991737768049, −3.19995102075144126385400170730, −2.82787637262431026122440533936, −1.92835007949537262955456323689, −0.36168034830975287574110869452,
0.36168034830975287574110869452, 1.92835007949537262955456323689, 2.82787637262431026122440533936, 3.19995102075144126385400170730, 4.51632642511709227991737768049, 5.09927947872366767473327366525, 5.52185229862619635051990862026, 6.57006288553514176971124892132, 7.14922111252467631052199215501, 7.945297023190132210941362505631