| L(s) = 1 | − 15.0·2-s − 285.·4-s − 1.87e3·5-s − 7.04e3·7-s + 1.20e4·8-s + 2.82e4·10-s + 2.47e3·11-s + 1.03e5·13-s + 1.06e5·14-s − 3.49e4·16-s + 5.29e5·17-s − 4.31e5·19-s + 5.35e5·20-s − 3.73e4·22-s + 1.00e5·23-s + 1.56e6·25-s − 1.55e6·26-s + 2.00e6·28-s + 7.09e6·29-s − 3.27e6·31-s − 5.62e6·32-s − 7.97e6·34-s + 1.32e7·35-s − 4.41e6·37-s + 6.50e6·38-s − 2.25e7·40-s − 1.81e7·41-s + ⋯ |
| L(s) = 1 | − 0.665·2-s − 0.556·4-s − 1.34·5-s − 1.10·7-s + 1.03·8-s + 0.894·10-s + 0.0510·11-s + 1.00·13-s + 0.737·14-s − 0.133·16-s + 1.53·17-s − 0.760·19-s + 0.747·20-s − 0.0339·22-s + 0.0749·23-s + 0.803·25-s − 0.666·26-s + 0.617·28-s + 1.86·29-s − 0.636·31-s − 0.947·32-s − 1.02·34-s + 1.48·35-s − 0.387·37-s + 0.506·38-s − 1.39·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4868346305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4868346305\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 + 15.0T + 512T^{2} \) |
| 5 | \( 1 + 1.87e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.04e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.47e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.03e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.00e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.09e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.41e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.81e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 5.52e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.53e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.01e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.09e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.07e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.81e8T + 7.60e17T^{2} \) |
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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
−9.847683021086806623299474100108, −8.679843435188212605635688214770, −8.218263729123623129840889372841, −7.25815393081520945806634707186, −6.23969436233856420395905122482, −4.87491861748512228919651132815, −3.79009244503468289138216468255, −3.23439428523708565038081929843, −1.30398703400478036456871720908, −0.36737640504117790908416739911,
0.36737640504117790908416739911, 1.30398703400478036456871720908, 3.23439428523708565038081929843, 3.79009244503468289138216468255, 4.87491861748512228919651132815, 6.23969436233856420395905122482, 7.25815393081520945806634707186, 8.218263729123623129840889372841, 8.679843435188212605635688214770, 9.847683021086806623299474100108
