| L(s) = 1 | + 89.5i·3-s + 1.19e3i·7-s − 5.82e3·9-s + 6.18e3·11-s + 7.65e3i·13-s + 2.97e4i·17-s − 1.02e4·19-s − 1.06e5·21-s + 4.53e4i·23-s − 3.25e5i·27-s − 1.29e5·29-s − 1.46e5·31-s + 5.53e5i·33-s + 1.54e4i·37-s − 6.85e5·39-s + ⋯ |
| L(s) = 1 | + 1.91i·3-s + 1.31i·7-s − 2.66·9-s + 1.40·11-s + 0.966i·13-s + 1.47i·17-s − 0.341·19-s − 2.51·21-s + 0.777i·23-s − 3.18i·27-s − 0.983·29-s − 0.880·31-s + 2.68i·33-s + 0.0502i·37-s − 1.85·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.675572100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675572100\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 89.5iT - 2.18e3T^{2} \) |
| 7 | \( 1 - 1.19e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 6.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.65e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.97e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.02e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.53e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.29e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.46e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.54e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.73e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.53e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.14e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 7.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.25e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.04e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 7.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.03e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 6.75e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.74e5iT - 8.07e13T^{2} \) |
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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
−11.49341638131805587041286177357, −10.95096817404246175922164049868, −9.509254417476570336503685608691, −9.259283704952042579695284406752, −8.362240396469841777769957046578, −6.28837985276690116617006553189, −5.50145790640471388813418224259, −4.24586645950244107473115978882, −3.55308740472342209195376665094, −2.00647810383186375285608207577,
0.48257013906717473006626664706, 1.02306186432676175227631380234, 2.36019531924173591080818944951, 3.77252813908133380454118013276, 5.54836281948629796056133090128, 6.80673615413634007653966243818, 7.19225075618947949670320287931, 8.175123906069734972040574564354, 9.323895173678730219771276803776, 10.84583592627256181952099420596
