L(s) = 1 | + (0.647 + 0.762i)2-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (−0.994 + 0.108i)7-s + (−0.856 + 0.515i)8-s + (−0.976 + 0.214i)10-s + (−0.947 + 0.319i)11-s + (−0.267 − 0.963i)13-s + (−0.725 − 0.687i)14-s + (−0.947 − 0.319i)16-s + (0.994 + 0.108i)17-s + (0.0541 + 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.856 − 0.515i)22-s + (0.907 + 0.419i)23-s + ⋯ |
L(s) = 1 | + (0.647 + 0.762i)2-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (−0.994 + 0.108i)7-s + (−0.856 + 0.515i)8-s + (−0.976 + 0.214i)10-s + (−0.947 + 0.319i)11-s + (−0.267 − 0.963i)13-s + (−0.725 − 0.687i)14-s + (−0.947 − 0.319i)16-s + (0.994 + 0.108i)17-s + (0.0541 + 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.856 − 0.515i)22-s + (0.907 + 0.419i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06647327469 + 0.9604672547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06647327469 + 0.9604672547i\) |
\(L(1)\) |
\(\approx\) |
\(0.7235098360 + 0.7361460856i\) |
\(L(1)\) |
\(\approx\) |
\(0.7235098360 + 0.7361460856i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 \) |
good | 2 | \( 1 + (0.647 + 0.762i)T \) |
| 5 | \( 1 + (-0.468 + 0.883i)T \) |
| 7 | \( 1 + (-0.994 + 0.108i)T \) |
| 11 | \( 1 + (-0.947 + 0.319i)T \) |
| 13 | \( 1 + (-0.267 - 0.963i)T \) |
| 17 | \( 1 + (0.994 + 0.108i)T \) |
| 19 | \( 1 + (0.0541 + 0.998i)T \) |
| 23 | \( 1 + (0.907 + 0.419i)T \) |
| 29 | \( 1 + (-0.647 + 0.762i)T \) |
| 31 | \( 1 + (-0.0541 + 0.998i)T \) |
| 37 | \( 1 + (0.856 + 0.515i)T \) |
| 41 | \( 1 + (-0.907 + 0.419i)T \) |
| 43 | \( 1 + (0.947 + 0.319i)T \) |
| 47 | \( 1 + (0.468 + 0.883i)T \) |
| 53 | \( 1 + (-0.976 - 0.214i)T \) |
| 61 | \( 1 + (-0.647 - 0.762i)T \) |
| 67 | \( 1 + (0.856 - 0.515i)T \) |
| 71 | \( 1 + (-0.468 - 0.883i)T \) |
| 73 | \( 1 + (0.725 + 0.687i)T \) |
| 79 | \( 1 + (0.796 + 0.605i)T \) |
| 83 | \( 1 + (-0.370 + 0.928i)T \) |
| 89 | \( 1 + (0.647 - 0.762i)T \) |
| 97 | \( 1 + (0.725 - 0.687i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.11415988099194240658467708446, −26.06653337522616008820021783515, −24.7063988291003358354803438503, −23.76481445530831586274821232199, −23.19115158648692357118391151287, −22.04554881532194313485797310275, −21.05687910616763248980885595728, −20.31790256234593619903569242191, −19.20765331096563822622365870138, −18.75369985229552668366150614235, −16.93701544100726559673471436700, −16.018398007407970630839891462863, −15.08194307812606851177602331077, −13.62730244187822831231640920859, −12.974266467781323384947521833482, −12.07323705765562392087887930630, −11.05272008119534632718264243540, −9.76915242997181479669688128790, −8.96251387347481626384293433947, −7.33880335361407792334545923424, −5.87873332787704165486194767000, −4.80671046076478552724899611446, −3.70261709330229062819427588437, −2.46415245603388500230125674098, −0.61401633309432513430863581370,
2.90095280743092263935957442628, 3.48391634056884322401427145349, 5.153444295481801117600313697592, 6.18554758344366703701453082091, 7.33112482630818331423548621735, 8.026927905284433854773445874218, 9.69761988194653919121364475962, 10.80731687269709927335542243270, 12.314095982053233688306614883699, 12.91006617840314338960282505618, 14.210368649395517281369552064846, 15.13741280915531115919357221127, 15.82711174964999509767226533832, 16.83439859315335814976439095226, 18.12375547809218371048219544791, 18.913914106051239167961426595144, 20.23289916503402475873682290100, 21.39945234826487790857040393564, 22.458333131009433295776109175122, 23.02996409919370874110265686303, 23.72815905580367404244279116021, 25.33258355172385618379831214112, 25.5912340126731173993432284484, 26.72026611461165497397381765686, 27.47723653904618597681162708590