L(s) = 1 | + (−1.39 + 0.238i)2-s + i·3-s + (1.88 − 0.664i)4-s − 4.40i·5-s + (−0.238 − 1.39i)6-s + 4.93·7-s + (−2.47 + 1.37i)8-s − 9-s + (1.05 + 6.14i)10-s + 0.318i·11-s + (0.664 + 1.88i)12-s − 3.58i·13-s + (−6.88 + 1.17i)14-s + 4.40·15-s + (3.11 − 2.50i)16-s − 2.25·17-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.168i)2-s + 0.577i·3-s + (0.943 − 0.332i)4-s − 1.97i·5-s + (−0.0973 − 0.569i)6-s + 1.86·7-s + (−0.873 + 0.486i)8-s − 0.333·9-s + (0.332 + 1.94i)10-s + 0.0960i·11-s + (0.191 + 0.544i)12-s − 0.995i·13-s + (−1.83 + 0.314i)14-s + 1.13·15-s + (0.779 − 0.626i)16-s − 0.546·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.906057 - 0.532532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.906057 - 0.532532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.238i)T \) |
| 3 | \( 1 - iT \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 4.40iT - 5T^{2} \) |
| 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 - 0.318iT - 11T^{2} \) |
| 13 | \( 1 + 3.58iT - 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 + 1.36iT - 19T^{2} \) |
| 29 | \( 1 - 1.30iT - 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 7.84iT - 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 9.90iT - 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 + 0.116iT - 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 8.74iT - 61T^{2} \) |
| 67 | \( 1 + 6.96iT - 67T^{2} \) |
| 71 | \( 1 - 3.93T + 71T^{2} \) |
| 73 | \( 1 - 0.227T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 3.29iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−10.65889375819149661380732623534, −9.404570582702183577635297787901, −8.899690762607944931025059942122, −8.112871880538201104754669631597, −7.65335859626603714365534350442, −5.71315306232784713436924787677, −5.13061635132584801727737112420, −4.24790181859662372851824187088, −2.05322533685589677781815972915, −0.869583258696784807257894728226,
1.78260221075951823525107820410, 2.50439217268032452297289499765, 3.97456546194801304299698307372, 5.80908937987921726310053933513, 6.80738624005600038184859894687, 7.40551857849986851489848881028, 8.094426225587254667578372361930, 9.085775041370951365954135973709, 10.34793580933763116919621314803, 10.93776851017061200449054374986