| L(s) = 1 | + 9.82·2-s − 9·3-s + 64.4·4-s + 77.2·5-s − 88.3·6-s − 23.9·7-s + 318.·8-s + 81·9-s + 758.·10-s + 145.·11-s − 580.·12-s + 635.·13-s − 235.·14-s − 694.·15-s + 1.06e3·16-s − 1.15e3·17-s + 795.·18-s + 90.4·19-s + 4.97e3·20-s + 215.·21-s + 1.43e3·22-s + 1.48e3·23-s − 2.86e3·24-s + 2.83e3·25-s + 6.24e3·26-s − 729·27-s − 1.54e3·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.01·4-s + 1.38·5-s − 1.00·6-s − 0.184·7-s + 1.76·8-s + 0.333·9-s + 2.39·10-s + 0.363·11-s − 1.16·12-s + 1.04·13-s − 0.321·14-s − 0.797·15-s + 1.04·16-s − 0.971·17-s + 0.578·18-s + 0.0574·19-s + 2.78·20-s + 0.106·21-s + 0.630·22-s + 0.583·23-s − 1.01·24-s + 0.907·25-s + 1.81·26-s − 0.192·27-s − 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.027125838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.027125838\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 9.82T + 32T^{2} \) |
| 5 | \( 1 - 77.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 23.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 145.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 635.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 90.4T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.06e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.74e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 6.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.79e4T + 8.58e9T^{2} \) |
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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
−12.06250713082407886120903450612, −11.08374885307063197479065827930, −10.21114251865598657728594133770, −8.850595911821867782853072692791, −6.76122324980814209154631294642, −6.26377397732042430052395000714, −5.35884806295060782471395813348, −4.31735911155321279922574474131, −2.86278985379174049968612595519, −1.51410512353981281761610422527,
1.51410512353981281761610422527, 2.86278985379174049968612595519, 4.31735911155321279922574474131, 5.35884806295060782471395813348, 6.26377397732042430052395000714, 6.76122324980814209154631294642, 8.850595911821867782853072692791, 10.21114251865598657728594133770, 11.08374885307063197479065827930, 12.06250713082407886120903450612
