| L(s) = 1 | − 1.28·2-s − 126.·4-s + 125·5-s − 538.·7-s + 326.·8-s − 160.·10-s + 1.21e3·11-s + 7.07e3·13-s + 690.·14-s + 1.57e4·16-s + 3.34e3·17-s + 2.21e4·19-s − 1.57e4·20-s − 1.55e3·22-s + 5.85e4·23-s + 1.56e4·25-s − 9.06e3·26-s + 6.80e4·28-s + 2.06e5·29-s + 1.77e5·31-s − 6.19e4·32-s − 4.29e3·34-s − 6.72e4·35-s − 2.84e5·37-s − 2.84e4·38-s + 4.07e4·40-s − 6.27e5·41-s + ⋯ |
| L(s) = 1 | − 0.113·2-s − 0.987·4-s + 0.447·5-s − 0.593·7-s + 0.225·8-s − 0.0506·10-s + 0.275·11-s + 0.892·13-s + 0.0672·14-s + 0.961·16-s + 0.165·17-s + 0.741·19-s − 0.441·20-s − 0.0311·22-s + 1.00·23-s + 0.199·25-s − 0.101·26-s + 0.585·28-s + 1.57·29-s + 1.07·31-s − 0.334·32-s − 0.0187·34-s − 0.265·35-s − 0.922·37-s − 0.0840·38-s + 0.100·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.438089847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438089847\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 2 | \( 1 + 1.28T + 128T^{2} \) |
| 7 | \( 1 + 538.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.07e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.34e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.42e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.59e6T + 8.07e13T^{2} \) |
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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
−14.00493737382901278011603018360, −13.39297133610143059644540145269, −12.15185124406984080145182921076, −10.45154286020594823189006120946, −9.416490350687852440185497505153, −8.369841317355816189714843271932, −6.53761132291264760759620629903, −5.04126837951953862533885090518, −3.37492276241368616820483871399, −0.970534731658867784971968568513,
0.970534731658867784971968568513, 3.37492276241368616820483871399, 5.04126837951953862533885090518, 6.53761132291264760759620629903, 8.369841317355816189714843271932, 9.416490350687852440185497505153, 10.45154286020594823189006120946, 12.15185124406984080145182921076, 13.39297133610143059644540145269, 14.00493737382901278011603018360
