| L(s) = 1 | − 2-s + 1.06·3-s + 4-s + 0.442·5-s − 1.06·6-s − 1.23·7-s − 8-s − 1.85·9-s − 0.442·10-s − 4.94·11-s + 1.06·12-s − 2.90·13-s + 1.23·14-s + 0.472·15-s + 16-s + 7.57·17-s + 1.85·18-s + 3.72·19-s + 0.442·20-s − 1.32·21-s + 4.94·22-s − 0.526·23-s − 1.06·24-s − 4.80·25-s + 2.90·26-s − 5.18·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.616·3-s + 0.5·4-s + 0.197·5-s − 0.435·6-s − 0.468·7-s − 0.353·8-s − 0.619·9-s − 0.139·10-s − 1.49·11-s + 0.308·12-s − 0.806·13-s + 0.330·14-s + 0.121·15-s + 0.250·16-s + 1.83·17-s + 0.438·18-s + 0.855·19-s + 0.0988·20-s − 0.288·21-s + 1.05·22-s − 0.109·23-s − 0.217·24-s − 0.960·25-s + 0.570·26-s − 0.998·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216136352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216136352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 0.442T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 - 7.57T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 23 | \( 1 + 0.526T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 9.22T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 0.945T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.45T + 79T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 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
−8.231317113549502414795344475074, −7.79447913700748032487831116917, −7.46075441328875642865674180401, −6.15622899735708613998530997254, −5.64702105867520115630629377652, −4.81089247321543603760499285574, −3.35817659978097352119762262170, −2.89964083744799240046182974869, −2.10973248413773254074847796631, −0.65035993521616334432877121499,
0.65035993521616334432877121499, 2.10973248413773254074847796631, 2.89964083744799240046182974869, 3.35817659978097352119762262170, 4.81089247321543603760499285574, 5.64702105867520115630629377652, 6.15622899735708613998530997254, 7.46075441328875642865674180401, 7.79447913700748032487831116917, 8.231317113549502414795344475074
