| L(s) = 1 | − 17.2·2-s − 81·3-s − 215.·4-s + 470.·5-s + 1.39e3·6-s − 5.41e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 8.10e3·10-s − 6.63e4·11-s + 1.74e4·12-s − 1.54e5·13-s + 9.31e4·14-s − 3.81e4·15-s − 1.05e5·16-s + 4.78e5·17-s − 1.12e5·18-s + 2.50e4·19-s − 1.01e5·20-s + 4.38e5·21-s + 1.14e6·22-s + 2.50e5·23-s − 1.01e6·24-s − 1.73e6·25-s + 2.66e6·26-s − 5.31e5·27-s + 1.16e6·28-s + ⋯ |
| L(s) = 1 | − 0.760·2-s − 0.577·3-s − 0.421·4-s + 0.336·5-s + 0.439·6-s − 0.852·7-s + 1.08·8-s + 0.333·9-s − 0.256·10-s − 1.36·11-s + 0.243·12-s − 1.50·13-s + 0.648·14-s − 0.194·15-s − 0.401·16-s + 1.38·17-s − 0.253·18-s + 0.0440·19-s − 0.141·20-s + 0.491·21-s + 1.04·22-s + 0.186·23-s − 0.624·24-s − 0.886·25-s + 1.14·26-s − 0.192·27-s + 0.358·28-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
| \(L(5)\) |
\(\approx\) |
\(0.1050969103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1050969103\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 17.2T + 512T^{2} \) |
| 5 | \( 1 - 470.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.41e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.63e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.54e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.78e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.50e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.50e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.66e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.81e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.23e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.87e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.96e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.03e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.36e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 8.61e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.81e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.97e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.72e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.76e8T + 7.60e17T^{2} \) |
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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
−10.49053166031987364197686031163, −10.05427330853200713125052269957, −9.290461195503873986577944404870, −7.88694752669060606269224983002, −7.16904924362967532118199302879, −5.61614519307517951385194819413, −4.90353621712583699146975747362, −3.26130352366531466833856116566, −1.75848335638355450904595992862, −0.18086211324099767883527556936,
0.18086211324099767883527556936, 1.75848335638355450904595992862, 3.26130352366531466833856116566, 4.90353621712583699146975747362, 5.61614519307517951385194819413, 7.16904924362967532118199302879, 7.88694752669060606269224983002, 9.290461195503873986577944404870, 10.05427330853200713125052269957, 10.49053166031987364197686031163
