| L(s) = 1 | + 19.3·2-s + 50.8·3-s + 245.·4-s − 408.·5-s + 981.·6-s + 1.74e3·7-s + 2.26e3·8-s + 395.·9-s − 7.88e3·10-s − 2.84e3·11-s + 1.24e4·12-s − 2.49e3·13-s + 3.37e4·14-s − 2.07e4·15-s + 1.22e4·16-s − 2.88e4·17-s + 7.63e3·18-s + 3.83e4·19-s − 9.99e4·20-s + 8.87e4·21-s − 5.49e4·22-s − 4.20e4·23-s + 1.14e5·24-s + 8.84e4·25-s − 4.81e4·26-s − 9.10e4·27-s + 4.27e5·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.08·3-s + 1.91·4-s − 1.46·5-s + 1.85·6-s + 1.92·7-s + 1.56·8-s + 0.180·9-s − 2.49·10-s − 0.643·11-s + 2.08·12-s − 0.314·13-s + 3.28·14-s − 1.58·15-s + 0.750·16-s − 1.42·17-s + 0.308·18-s + 1.28·19-s − 2.79·20-s + 2.09·21-s − 1.09·22-s − 0.719·23-s + 1.69·24-s + 1.13·25-s − 0.537·26-s − 0.890·27-s + 3.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.103455379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.103455379\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.43e4T \) |
| good | 2 | \( 1 - 19.3T + 128T^{2} \) |
| 3 | \( 1 - 50.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 408.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.74e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.84e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.49e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.88e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.83e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.20e4T + 3.40e9T^{2} \) |
| 31 | \( 1 - 1.33e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.26e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.25e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.42e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.97e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.03e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.82e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.08e7T + 8.07e13T^{2} \) |
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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
−15.08228993287244395094063812919, −14.40567712358224709526742526884, −13.41617845701687667706026352273, −11.80008198391378798002292339166, −11.25631053107951349832587300508, −8.325539002852116211070353170704, −7.49681923028863403521970577384, −4.99769745279528697578740955019, −3.96188008194860952727573027447, −2.41578342546966125766797219275,
2.41578342546966125766797219275, 3.96188008194860952727573027447, 4.99769745279528697578740955019, 7.49681923028863403521970577384, 8.325539002852116211070353170704, 11.25631053107951349832587300508, 11.80008198391378798002292339166, 13.41617845701687667706026352273, 14.40567712358224709526742526884, 15.08228993287244395094063812919
