L(s) = 1 | + (−0.977 − 0.212i)2-s + (0.0487 + 0.998i)3-s + (0.909 + 0.416i)4-s + (−0.184 + 0.982i)5-s + (0.165 − 0.986i)6-s + (−0.145 − 0.989i)7-s + (−0.799 − 0.600i)8-s + (−0.995 + 0.0974i)9-s + (0.389 − 0.921i)10-s + (0.316 + 0.948i)11-s + (−0.371 + 0.928i)12-s + (0.316 − 0.948i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (0.653 + 0.756i)16-s + (0.353 + 0.935i)17-s + ⋯ |
L(s) = 1 | + (−0.977 − 0.212i)2-s + (0.0487 + 0.998i)3-s + (0.909 + 0.416i)4-s + (−0.184 + 0.982i)5-s + (0.165 − 0.986i)6-s + (−0.145 − 0.989i)7-s + (−0.799 − 0.600i)8-s + (−0.995 + 0.0974i)9-s + (0.389 − 0.921i)10-s + (0.316 + 0.948i)11-s + (−0.371 + 0.928i)12-s + (0.316 − 0.948i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (0.653 + 0.756i)16-s + (0.353 + 0.935i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07348159829 + 0.5806751634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07348159829 + 0.5806751634i\) |
\(L(1)\) |
\(\approx\) |
\(0.5486833821 + 0.2960856785i\) |
\(L(1)\) |
\(\approx\) |
\(0.5486833821 + 0.2960856785i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.977 - 0.212i)T \) |
| 3 | \( 1 + (0.0487 + 0.998i)T \) |
| 5 | \( 1 + (-0.184 + 0.982i)T \) |
| 7 | \( 1 + (-0.145 - 0.989i)T \) |
| 11 | \( 1 + (0.316 + 0.948i)T \) |
| 13 | \( 1 + (0.316 - 0.948i)T \) |
| 17 | \( 1 + (0.353 + 0.935i)T \) |
| 19 | \( 1 + (0.811 + 0.584i)T \) |
| 23 | \( 1 + (-0.999 + 0.0195i)T \) |
| 29 | \( 1 + (-0.696 + 0.717i)T \) |
| 31 | \( 1 + (0.763 - 0.646i)T \) |
| 37 | \( 1 + (0.938 - 0.344i)T \) |
| 41 | \( 1 + (-0.576 + 0.816i)T \) |
| 43 | \( 1 + (0.0876 + 0.996i)T \) |
| 47 | \( 1 + (-0.724 + 0.689i)T \) |
| 53 | \( 1 + (0.203 - 0.979i)T \) |
| 59 | \( 1 + (-0.638 + 0.769i)T \) |
| 61 | \( 1 + (-0.576 + 0.816i)T \) |
| 67 | \( 1 + (0.763 + 0.646i)T \) |
| 71 | \( 1 + (-0.977 + 0.212i)T \) |
| 73 | \( 1 + (0.203 + 0.979i)T \) |
| 79 | \( 1 + (0.241 - 0.970i)T \) |
| 83 | \( 1 + (0.924 + 0.380i)T \) |
| 89 | \( 1 + (0.763 + 0.646i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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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
−21.14895457922133662437654537440, −20.29181003074453861981116480845, −19.64439964230531050423976974107, −18.786793708856246388285212315730, −18.500626531484024721641059059028, −17.523196680561453405638842675660, −16.68443090819558115047495875103, −16.11498311632779484944758757231, −15.3450650885727656908280717378, −14.05907395889853843512438227199, −13.53298335690632838625585940910, −12.11594345304576019909575268216, −11.903622642726906918485702072868, −11.18376830628785057021466138604, −9.55618546779871452755828692504, −9.0641846676850079197090801429, −8.37131205956684784466474861321, −7.6807269098170688367742887165, −6.60424733272315263066670395842, −5.89862664470266383277716556445, −5.10753706520732180367992328522, −3.359550323974935127885151419758, −2.281474165415431073176137600774, −1.401427851679834432554455081971, −0.36558334011533195240234576245,
1.378308280399678498311486072988, 2.73446362535696163331804266812, 3.55126780448161512268683170941, 4.18266111136665580861432353834, 5.78298170300586161336819573374, 6.59432191158956575553416927720, 7.74420667527808076443316857441, 8.060427358692289409749480081439, 9.59573430014270999003732798432, 9.9723003756713501813601465210, 10.59873298424944467661162210228, 11.2803104509539950261186376267, 12.17583905284421813510156783263, 13.368133590178509568383746666690, 14.70900688717647750478572384495, 14.89406556720300256554132226497, 16.01772088789038758754781297475, 16.54184860460680078299040983581, 17.5846939228191776249046001567, 17.926535758761073159313335894410, 19.067636398117676800769300766996, 20.03063557164167467429443705069, 20.18929369422614497899090876485, 21.1457858266251576866475435906, 22.073906452487939670703234405603