L(s) = 1 | + 81i·3-s − 1.65e3i·5-s + 1.13e4·7-s − 6.56e3·9-s − 9.45e4i·11-s + 1.40e5i·13-s + 1.34e5·15-s − 5.30e5·17-s − 3.87e5i·19-s + 9.20e5i·21-s + 2.36e6·23-s − 7.87e5·25-s − 5.31e5i·27-s − 2.70e6i·29-s + 5.18e6·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.18i·5-s + 1.78·7-s − 0.333·9-s − 1.94i·11-s + 1.36i·13-s + 0.683·15-s − 1.54·17-s − 0.682i·19-s + 1.03i·21-s + 1.75·23-s − 0.403·25-s − 0.192i·27-s − 0.710i·29-s + 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.722801183\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.722801183\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
good | 5 | \( 1 + 1.65e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 1.13e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 9.45e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.40e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 5.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.87e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 2.36e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.70e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 5.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.42e5iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.65e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.01e6iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 9.71e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 7.23e6iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 2.34e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.62e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.01e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.48e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 9.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.09e8T + 7.60e17T^{2} \) |
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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
−9.099101972301053771903012595782, −8.804416574918000527713017034206, −8.182039266615549393240558948623, −6.72906415591829276351612506983, −5.44169310900397595268300241823, −4.72686463474215589918107121206, −4.14038623038402991313322548065, −2.53950719611878852975291890971, −1.31921339155735563815116127243, −0.51773319051972491620878935797,
1.14959344896462001012048123787, 2.07097153677744434388831862949, 2.85427096881113087489005183209, 4.45677387637289202672790819094, 5.17876365886436282210502149815, 6.58010336901169639908219398499, 7.33892011846494224880289484703, 7.906870319150127963920335871005, 8.999885405405963698684029883838, 10.42446869817449532104997392148