| L(s) = 1 | + 8·2-s − 92.4·3-s + 64·4-s − 426.·5-s − 739.·6-s + 875.·7-s + 512·8-s + 6.36e3·9-s − 3.41e3·10-s − 1.92e3·11-s − 5.91e3·12-s + 6.35e3·13-s + 7.00e3·14-s + 3.94e4·15-s + 4.09e3·16-s − 1.33e4·17-s + 5.09e4·18-s + 6.85e3·19-s − 2.72e4·20-s − 8.09e4·21-s − 1.53e4·22-s + 5.49e4·23-s − 4.73e4·24-s + 1.03e5·25-s + 5.08e4·26-s − 3.86e5·27-s + 5.60e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.97·3-s + 0.5·4-s − 1.52·5-s − 1.39·6-s + 0.964·7-s + 0.353·8-s + 2.91·9-s − 1.07·10-s − 0.436·11-s − 0.988·12-s + 0.801·13-s + 0.682·14-s + 3.01·15-s + 0.250·16-s − 0.659·17-s + 2.05·18-s + 0.229·19-s − 0.762·20-s − 1.90·21-s − 0.308·22-s + 0.941·23-s − 0.699·24-s + 1.32·25-s + 0.567·26-s − 3.77·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.159028814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159028814\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 + 92.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 426.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 875.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.92e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.35e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.33e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 5.49e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.17e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.03e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.18e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.34e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.24e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.87e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.14e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.72e6T + 8.07e13T^{2} \) |
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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
−15.27192415567624564040689558842, −13.20095426412082243135132731038, −11.98524202329443079747875574678, −11.39268626731434536041737620665, −10.69259498468142845607361922200, −7.86795451400915874137527167136, −6.60870133113669648643913424612, −5.08163851258001102960551192924, −4.19889929609670336305552458801, −0.857478437018250913227287613304,
0.857478437018250913227287613304, 4.19889929609670336305552458801, 5.08163851258001102960551192924, 6.60870133113669648643913424612, 7.86795451400915874137527167136, 10.69259498468142845607361922200, 11.39268626731434536041737620665, 11.98524202329443079747875574678, 13.20095426412082243135132731038, 15.27192415567624564040689558842
