| L(s) = 1 | − 1.73i·3-s − 4·7-s − 2.99·9-s + 6.92i·13-s + (4 + 1.73i)19-s + 6.92i·21-s + 5·25-s + 5.19i·27-s + 10.3i·31-s − 6.92i·37-s + 11.9·39-s − 8·43-s + 9·49-s + (2.99 − 6.92i)57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s − 1.51·7-s − 0.999·9-s + 1.92i·13-s + (0.917 + 0.397i)19-s + 1.51i·21-s + 25-s + 0.999i·27-s + 1.86i·31-s − 1.13i·37-s + 1.92·39-s − 1.21·43-s + 1.28·49-s + (0.397 − 0.917i)57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.639832 + 0.420185i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.639832 + 0.420185i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.16761401018975576072128546912, −9.204770062142631447508928788500, −8.769787722509297637281614053890, −7.43590336353570720291134580689, −6.76865001662113966146381760546, −6.30212896485923721085145632104, −5.12799861132040569122094833053, −3.69832705049264906205962252807, −2.76664296321127246458074039600, −1.43059509108331842047903431619,
0.36983345952922080602492014411, 3.01966082202961335467321198093, 3.21295115422570410239520885970, 4.59908814655601506373740050463, 5.59665502603094405824843497903, 6.25752534180307998816673657423, 7.45556470841348436466321618004, 8.412089248664779769199288113932, 9.351770335484150192455132885692, 9.960644255468066515084109212125
