| L(s) = 1 | − 12.8·2-s + 81·3-s − 346.·4-s − 288.·5-s − 1.04e3·6-s − 1.11e4·7-s + 1.10e4·8-s + 6.56e3·9-s + 3.70e3·10-s − 5.35e3·11-s − 2.81e4·12-s − 1.19e4·13-s + 1.43e5·14-s − 2.33e4·15-s + 3.58e4·16-s − 2.53e5·17-s − 8.42e4·18-s − 3.68e5·19-s + 9.99e4·20-s − 9.03e5·21-s + 6.87e4·22-s − 1.43e6·23-s + 8.93e5·24-s − 1.87e6·25-s + 1.53e5·26-s + 5.31e5·27-s + 3.87e6·28-s + ⋯ |
| L(s) = 1 | − 0.567·2-s + 0.577·3-s − 0.677·4-s − 0.206·5-s − 0.327·6-s − 1.75·7-s + 0.952·8-s + 0.333·9-s + 0.117·10-s − 0.110·11-s − 0.391·12-s − 0.116·13-s + 0.997·14-s − 0.119·15-s + 0.136·16-s − 0.735·17-s − 0.189·18-s − 0.648·19-s + 0.139·20-s − 1.01·21-s + 0.0626·22-s − 1.06·23-s + 0.549·24-s − 0.957·25-s + 0.0659·26-s + 0.192·27-s + 1.19·28-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
| \(L(5)\) |
\(\approx\) |
\(0.3371057685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3371057685\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 12.8T + 512T^{2} \) |
| 5 | \( 1 + 288.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.11e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.35e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.19e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.68e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.52e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.80e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.18e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.28e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 9.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.80e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.83e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.87e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.30e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.65e8T + 7.60e17T^{2} \) |
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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
−10.58576884699469906785676099171, −9.742408346518968087680326705968, −9.157676422632193464233677170860, −8.159065239531232092533395545720, −7.10186969027604911573554729559, −5.92755998069749231608856788584, −4.27311709234337285828004765595, −3.43716657544367461228674620420, −2.02535742296626694591640008180, −0.28983805065045137118969105659,
0.28983805065045137118969105659, 2.02535742296626694591640008180, 3.43716657544367461228674620420, 4.27311709234337285828004765595, 5.92755998069749231608856788584, 7.10186969027604911573554729559, 8.159065239531232092533395545720, 9.157676422632193464233677170860, 9.742408346518968087680326705968, 10.58576884699469906785676099171
