| L(s) = 1 | − 4.85·2-s + 15.6·4-s − 14.8·5-s + 29.6·7-s − 36.9·8-s + 72.2·10-s + 60.0·11-s − 144.·14-s + 54.7·16-s − 49.0·17-s + 96.7·19-s − 232.·20-s − 291.·22-s + 42.4·23-s + 96.1·25-s + 462.·28-s + 176.·29-s + 269.·31-s + 29.6·32-s + 238.·34-s − 441.·35-s + 186.·37-s − 470.·38-s + 549.·40-s + 52.7·41-s + 527.·43-s + 936.·44-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.95·4-s − 1.33·5-s + 1.60·7-s − 1.63·8-s + 2.28·10-s + 1.64·11-s − 2.75·14-s + 0.855·16-s − 0.700·17-s + 1.16·19-s − 2.59·20-s − 2.82·22-s + 0.384·23-s + 0.769·25-s + 3.12·28-s + 1.13·29-s + 1.56·31-s + 0.164·32-s + 1.20·34-s − 2.13·35-s + 0.829·37-s − 2.00·38-s + 2.17·40-s + 0.200·41-s + 1.87·43-s + 3.21·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.150803973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150803973\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.85T + 8T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 - 29.6T + 343T^{2} \) |
| 11 | \( 1 - 60.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 49.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 42.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 269.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 186.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 52.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 527.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 61.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 595.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 415.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 640.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 551.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 431.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 998.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 461.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 261.T + 9.12e5T^{2} \) |
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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
−8.952544510134988657914202612305, −8.296100174836674411417461788966, −7.77998882457782699012560896122, −7.13715648437478297388050122807, −6.29041477938238067073536314582, −4.73363398090993673881468731998, −4.06573808155098949188177524987, −2.64478977711068259347630622509, −1.31100774052871216212335564989, −0.814383823201349268331132513988,
0.814383823201349268331132513988, 1.31100774052871216212335564989, 2.64478977711068259347630622509, 4.06573808155098949188177524987, 4.73363398090993673881468731998, 6.29041477938238067073536314582, 7.13715648437478297388050122807, 7.77998882457782699012560896122, 8.296100174836674411417461788966, 8.952544510134988657914202612305
