| L(s) = 1 | − 0.732·3-s − 2.73i·7-s − 2.46·9-s + 2i·11-s − 3.46·13-s + 3.46i·17-s + 7.46i·19-s + 2i·21-s − 4.19i·23-s + 4·27-s + 6.92i·29-s − 1.46·31-s − 1.46i·33-s − 2·37-s + 2.53·39-s + ⋯ |
| L(s) = 1 | − 0.422·3-s − 1.03i·7-s − 0.821·9-s + 0.603i·11-s − 0.960·13-s + 0.840i·17-s + 1.71i·19-s + 0.436i·21-s − 0.874i·23-s + 0.769·27-s + 1.28i·29-s − 0.262·31-s − 0.254i·33-s − 0.328·37-s + 0.406·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.168011 + 0.373536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.168011 + 0.373536i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 + 4.19iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 - 6.73iT - 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 4.92iT - 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 - 0.535iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39iT - 97T^{2} \) |
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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
−10.33971338155455910654517772829, −10.14278536454825134496276268351, −8.815802603217801937837244232209, −7.959631945086510105715149545360, −7.10814942907766504385392273905, −6.24051649265332756439272833205, −5.23137781273671308294036010667, −4.29686576740692511376344309883, −3.19875189008048115435963113536, −1.64427633098665513122087339943,
0.20530186332446194564206081423, 2.34239977120619844696625547455, 3.18200286231576992832809567057, 4.88407789672504954709828188624, 5.40453170544000197434787236152, 6.35806061421585888655390720022, 7.30400507093415134794035337227, 8.413916345956317663786644901892, 9.092544269166637624577617759526, 9.843151173190354857571567810624
