L(s) = 1 | + 2i·2-s − 2·4-s + 3i·5-s − i·7-s − 3·9-s − 6·10-s − 6i·11-s + 2·14-s − 4·16-s − 4·17-s − 6i·18-s − 5i·19-s − 6i·20-s + 12·22-s − 3·23-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 4-s + 1.34i·5-s − 0.377i·7-s − 9-s − 1.89·10-s − 1.80i·11-s + 0.534·14-s − 16-s − 0.970·17-s − 1.41i·18-s − 1.14i·19-s − 1.34i·20-s + 2.55·22-s − 0.625·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 5iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 3iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 - 3iT - 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.216434654102498881900676986568, −8.647665992310408386362152389604, −7.79564503359671757395302267518, −7.08747754694164583198811242244, −6.14853283728876422680485728492, −6.00146053406771464552867517460, −4.73171856981058939699108533044, −3.38289976035022851334095946634, −2.58707040853750830679738532681, 0,
1.68179844572142180891700624983, 2.28058876041277896516478184491, 3.70358442199176084348655640008, 4.54102261865655065215199450947, 5.29015450905288561683676006858, 6.43564563033490143101557607390, 7.71968146717104930633286583149, 8.605054182995359466452401331806, 9.342165744759600001177971385751