| L(s) = 1 | − 38.0·2-s + 81·3-s + 939.·4-s − 310.·5-s − 3.08e3·6-s − 4.49e3·7-s − 1.62e4·8-s + 6.56e3·9-s + 1.18e4·10-s − 5.77e4·11-s + 7.60e4·12-s − 1.58e5·13-s + 1.71e5·14-s − 2.51e4·15-s + 1.39e5·16-s − 4.06e5·17-s − 2.49e5·18-s − 5.88e5·19-s − 2.91e5·20-s − 3.63e5·21-s + 2.20e6·22-s − 2.03e6·23-s − 1.31e6·24-s − 1.85e6·25-s + 6.02e6·26-s + 5.31e5·27-s − 4.21e6·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.83·4-s − 0.221·5-s − 0.971·6-s − 0.707·7-s − 1.40·8-s + 0.333·9-s + 0.373·10-s − 1.18·11-s + 1.05·12-s − 1.53·13-s + 1.19·14-s − 0.128·15-s + 0.530·16-s − 1.18·17-s − 0.561·18-s − 1.03·19-s − 0.407·20-s − 0.408·21-s + 2.00·22-s − 1.51·23-s − 0.810·24-s − 0.950·25-s + 2.58·26-s + 0.192·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.09298824051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09298824051\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 38.0T + 512T^{2} \) |
| 5 | \( 1 + 310.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.49e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.58e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.06e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.76e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.04e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.85e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.47e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.12e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.38e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.25e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.34e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.26e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.98e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.54e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.81e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.21e8T + 7.60e17T^{2} \) |
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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
−10.35521623430501592898865174908, −10.04941410590721485889094886486, −8.974867262487331109492901740266, −8.097522901530690439223590727263, −7.37239664184529203220695007751, −6.32042722582928489484429184713, −4.46642572932949885282914351365, −2.68543876446031308524211029722, −2.04783957312966108409223313617, −0.17282345261579597758141864504,
0.17282345261579597758141864504, 2.04783957312966108409223313617, 2.68543876446031308524211029722, 4.46642572932949885282914351365, 6.32042722582928489484429184713, 7.37239664184529203220695007751, 8.097522901530690439223590727263, 8.974867262487331109492901740266, 10.04941410590721485889094886486, 10.35521623430501592898865174908
