L(s) = 1 | − 2.95i·3-s + 362. i·5-s − 715.·7-s + 2.17e3·9-s − 7.29e3i·11-s + 1.03e4i·13-s + 1.07e3·15-s − 1.12e4·17-s − 2.66e4i·19-s + 2.11e3i·21-s − 2.35e4·23-s − 5.32e4·25-s − 1.29e4i·27-s − 4.41e4i·29-s − 3.11e5·31-s + ⋯ |
L(s) = 1 | − 0.0632i·3-s + 1.29i·5-s − 0.787·7-s + 0.995·9-s − 1.65i·11-s + 1.31i·13-s + 0.0820·15-s − 0.554·17-s − 0.892i·19-s + 0.0498i·21-s − 0.403·23-s − 0.681·25-s − 0.126i·27-s − 0.336i·29-s − 1.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.484068650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.484068650\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 2.95iT - 2.18e3T^{2} \) |
| 5 | \( 1 - 362. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 715.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.29e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.03e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.66e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 2.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.41e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.39e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 7.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.25e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 7.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.14e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.01e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 5.67e3iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.80e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.78e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.85e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.79e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 5.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.46e7T + 8.07e13T^{2} \) |
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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
−10.92660603136837245735461540624, −9.678699177465684839825289244051, −8.907154027641478993083200656625, −7.39269979395075542100092978391, −6.71163944275683312798233872365, −5.93710800083193659885683185791, −4.16978886227099154952922046237, −3.24807781964716934271518009678, −2.10540455298829751109933998421, −0.40946986239079721841197335021,
0.930324732779931117048550557065, 2.06294726467352233028406604128, 3.79294029951798148963095882851, 4.69657816004705118433897306615, 5.70366138466731868732678950976, 7.07893544534935079931713604909, 7.913498028282631166801907336757, 9.190393200226963461923613134738, 9.793805187300032222000606283807, 10.64745181684091181746563690542