L(s) = 1 | + (−0.990 − 0.136i)2-s + (−0.775 − 0.631i)3-s + (0.962 + 0.269i)4-s + (0.854 + 0.519i)5-s + (0.682 + 0.730i)6-s + (0.460 − 0.887i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (−0.775 − 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.576 − 0.816i)12-s + (0.203 − 0.979i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (0.854 + 0.519i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.136i)2-s + (−0.775 − 0.631i)3-s + (0.962 + 0.269i)4-s + (0.854 + 0.519i)5-s + (0.682 + 0.730i)6-s + (0.460 − 0.887i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (−0.775 − 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.576 − 0.816i)12-s + (0.203 − 0.979i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (0.854 + 0.519i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7934545217 - 0.4087947829i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7934545217 - 0.4087947829i\) |
\(L(1)\) |
\(\approx\) |
\(0.6843676742 - 0.1730330744i\) |
\(L(1)\) |
\(\approx\) |
\(0.6843676742 - 0.1730330744i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.990 - 0.136i)T \) |
| 3 | \( 1 + (-0.775 - 0.631i)T \) |
| 5 | \( 1 + (0.854 + 0.519i)T \) |
| 7 | \( 1 + (0.460 - 0.887i)T \) |
| 11 | \( 1 + (0.203 + 0.979i)T \) |
| 13 | \( 1 + (0.203 - 0.979i)T \) |
| 17 | \( 1 + (-0.334 + 0.942i)T \) |
| 19 | \( 1 + (-0.775 - 0.631i)T \) |
| 23 | \( 1 + (0.962 + 0.269i)T \) |
| 29 | \( 1 + (0.203 - 0.979i)T \) |
| 31 | \( 1 + (-0.917 - 0.398i)T \) |
| 37 | \( 1 + (0.203 - 0.979i)T \) |
| 41 | \( 1 + (0.682 + 0.730i)T \) |
| 43 | \( 1 + (-0.334 - 0.942i)T \) |
| 47 | \( 1 + (-0.334 - 0.942i)T \) |
| 53 | \( 1 + (0.962 + 0.269i)T \) |
| 59 | \( 1 + (0.962 - 0.269i)T \) |
| 61 | \( 1 + (0.682 + 0.730i)T \) |
| 67 | \( 1 + (-0.917 + 0.398i)T \) |
| 71 | \( 1 + (-0.990 + 0.136i)T \) |
| 73 | \( 1 + (0.962 - 0.269i)T \) |
| 79 | \( 1 + (0.962 - 0.269i)T \) |
| 83 | \( 1 + (0.682 + 0.730i)T \) |
| 89 | \( 1 + (-0.917 + 0.398i)T \) |
| 97 | \( 1 + T \) |
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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
−21.52886952057631505222471493379, −21.16484879686659720451910387988, −20.53514281962080915920256628422, −19.279369916731831102531859684553, −18.431833950528502838671240066157, −17.9657795740269296677845909066, −17.05553975350960492240206427813, −16.41812094835002176874938112780, −16.03902940751010225246518457252, −14.86270599064988550475616375991, −14.232543482158085357427462948469, −12.83591170873451692164077492880, −11.92561559557256809284112687775, −11.23962880323966670471785329915, −10.59706230798645169913175540513, −9.494528965406767956469177117593, −8.998684452824629071138943317185, −8.45005508540399196021530416007, −6.87628967825568156700381281617, −6.18085589575667332113092821230, −5.46014851296541047955573214739, −4.658602615221059185322912091605, −3.11338832202078716131744885654, −1.93588266398436451824068414874, −0.97980171784568902863206338903,
0.745875332722452916686597197834, 1.779649390359084979354075579068, 2.43068234605662523833812915450, 3.95119746591432117016216618078, 5.28978189747805283370294201668, 6.21252055467477061517783438276, 6.9932180377174859454893324308, 7.51247823073207546263755990981, 8.52683248739854891207651611536, 9.68843855505278927883906788664, 10.58526019086566587043262359404, 10.78606959963476109937813153584, 11.756672353384440945682736339522, 12.97651223311735202974728154737, 13.2230406086000877308107424753, 14.71173371580541268989035070223, 15.27596701379323081833149016244, 16.65828989021201844671832719294, 17.19585211338583592095528762579, 17.77332711613979808964031524677, 18.0772778213152552804784528294, 19.243583927317594065276224955541, 19.8056093217890035983952470677, 20.78852828240673455586176479449, 21.50644565219017203514577014215