L(s) = 1 | + (2 − i)5-s − 4i·7-s − 4·11-s − 4i·13-s + 6i·17-s + 4·19-s − 4i·23-s + (3 − 4i)25-s + 4·29-s + (−4 − 8i)35-s + 4i·37-s + 8·41-s + 12i·47-s − 9·49-s − 2i·53-s + ⋯ |
L(s) = 1 | + (0.894 − 0.447i)5-s − 1.51i·7-s − 1.20·11-s − 1.10i·13-s + 1.45i·17-s + 0.917·19-s − 0.834i·23-s + (0.600 − 0.800i)25-s + 0.742·29-s + (−0.676 − 1.35i)35-s + 0.657i·37-s + 1.24·41-s + 1.75i·47-s − 1.28·49-s − 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22379 - 0.756346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22379 - 0.756346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.83570633696793494852455427576, −10.43355746923322554872980265295, −9.711675974419078154307610084218, −8.299380074429026524925732610899, −7.66822164365308683845383546332, −6.39229219938449426721449446526, −5.38636321607616423847974741462, −4.32417740296710715670041851557, −2.85451344586925836438889587997, −1.05956648832118516673276067628,
2.14246318192079397400286099028, 2.98152439754895514848236179863, 5.03965615536919006739960089996, 5.63478898908345632250485287133, 6.75750391678551577095774735061, 7.82619478472493033762344294927, 9.245279456768092535170402034086, 9.435168377190757686636595233818, 10.70732716903024852763786465084, 11.65879579346267356258879884230