L(s) = 1 | + (0.618 + 0.786i)2-s + (−0.520 + 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (−0.992 + 0.118i)6-s + (0.771 − 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (−0.707 − 0.707i)12-s + (0.977 + 0.212i)14-s + (0.992 + 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯ |
L(s) = 1 | + (0.618 + 0.786i)2-s + (−0.520 + 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (−0.992 + 0.118i)6-s + (0.771 − 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (−0.707 − 0.707i)12-s + (0.977 + 0.212i)14-s + (0.992 + 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1998447002 + 0.4034252527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1998447002 + 0.4034252527i\) |
\(L(1)\) |
\(\approx\) |
\(0.7143357083 + 0.5457298236i\) |
\(L(1)\) |
\(\approx\) |
\(0.7143357083 + 0.5457298236i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.618 + 0.786i)T \) |
| 3 | \( 1 + (-0.520 + 0.853i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.771 - 0.636i)T \) |
| 11 | \( 1 + (0.0950 + 0.995i)T \) |
| 17 | \( 1 + (0.618 - 0.786i)T \) |
| 19 | \( 1 + (-0.952 - 0.304i)T \) |
| 23 | \( 1 + (-0.739 + 0.672i)T \) |
| 29 | \( 1 + (0.165 + 0.986i)T \) |
| 31 | \( 1 + (-0.977 - 0.212i)T \) |
| 37 | \( 1 + (-0.258 + 0.965i)T \) |
| 41 | \( 1 + (-0.999 + 0.0237i)T \) |
| 43 | \( 1 + (-0.165 + 0.986i)T \) |
| 47 | \( 1 + (-0.959 + 0.281i)T \) |
| 53 | \( 1 + (-0.281 + 0.959i)T \) |
| 59 | \( 1 + (-0.853 + 0.520i)T \) |
| 61 | \( 1 + (-0.828 - 0.560i)T \) |
| 67 | \( 1 + (0.971 - 0.235i)T \) |
| 71 | \( 1 + (-0.580 - 0.814i)T \) |
| 73 | \( 1 + (-0.540 - 0.841i)T \) |
| 79 | \( 1 + (0.540 + 0.841i)T \) |
| 83 | \( 1 + (-0.599 - 0.800i)T \) |
| 97 | \( 1 + (-0.0950 + 0.995i)T \) |
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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
−20.97625552201258975976124294126, −19.80874730785455277657529762549, −19.09619360317226148860267874552, −18.68723279337206518478403004895, −18.077733412906150412625632383460, −17.13874753347587150664882684879, −16.027050495124324185650688669275, −14.9790959020001543142539014415, −14.40157890441786466571507350483, −13.77312833489963754829330516667, −12.71679904647893200002264213089, −12.09072841846364954937563621402, −11.4022937691955449206037139627, −10.845556234597045728949690001599, −10.15431577616300961036935023883, −8.590126877573993196931052620617, −8.06367601710325018184830279791, −6.81084808445984341699586472771, −6.00819853594357721423077668737, −5.503225340458374699563557652394, −4.26314293537035998633265750466, −3.331059431990759067313225188094, −2.28273341524059452492626168008, −1.65836291215662961759616916462, −0.152727053609296844839855283325,
1.53568026597019196125979714174, 3.27257368283518699752345858545, 4.14383181334581950589648021411, 4.80952214037434501654803774900, 5.165762294935666556851269901696, 6.34089169252630716013310346225, 7.34662480688759034218574564786, 8.05807320980136980584503223386, 9.002597244102272520909178666051, 9.75685986702115330251927334315, 10.90100454104416834747121101207, 11.79127226272493023086900157378, 12.29720032245959963002213561023, 13.23655867910032393392950727557, 14.217351562247636537405939868062, 14.95351180217256755818452739037, 15.54846010944395200803562133942, 16.4363878182557011522245531165, 16.87879725562356013545832567302, 17.560471941309428560136355726393, 18.260863409572509346413997436561, 19.99929290851287238877464873265, 20.38339186214227392286041245734, 21.14610393677389187232325783696, 21.75841506776597211273774885605