L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (0.433 − 0.900i)8-s + (0.974 − 0.222i)10-s + (−0.433 − 0.900i)11-s + (0.900 − 0.433i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s + i·17-s + (−0.974 + 0.222i)19-s + (−0.900 − 0.433i)20-s + (−0.222 + 0.974i)22-s + (−0.222 − 0.974i)25-s + (−0.974 − 0.222i)26-s + ⋯ |
L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (0.433 − 0.900i)8-s + (0.974 − 0.222i)10-s + (−0.433 − 0.900i)11-s + (0.900 − 0.433i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s + i·17-s + (−0.974 + 0.222i)19-s + (−0.900 − 0.433i)20-s + (−0.222 + 0.974i)22-s + (−0.222 − 0.974i)25-s + (−0.974 − 0.222i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07978862239 - 0.6201128068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07978862239 - 0.6201128068i\) |
\(L(1)\) |
\(\approx\) |
\(0.6080021059 - 0.2043928911i\) |
\(L(1)\) |
\(\approx\) |
\(0.6080021059 - 0.2043928911i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
| 11 | \( 1 + (-0.433 - 0.900i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.974 + 0.222i)T \) |
| 31 | \( 1 + (0.781 + 0.623i)T \) |
| 37 | \( 1 + (0.433 - 0.900i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.781 - 0.623i)T \) |
| 47 | \( 1 + (-0.433 - 0.900i)T \) |
| 53 | \( 1 + (0.623 - 0.781i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.974 + 0.222i)T \) |
| 67 | \( 1 + (-0.900 - 0.433i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.781 - 0.623i)T \) |
| 79 | \( 1 + (-0.433 + 0.900i)T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.781 + 0.623i)T \) |
| 97 | \( 1 + (0.974 - 0.222i)T \) |
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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
−20.04930661834381817983073140644, −19.08439886600593412871440328252, −18.70103330426332331061588175229, −17.88313979576737548652844146072, −17.23428051204149657052096111510, −16.36940425630725529235380230821, −15.70646089386088320219321739170, −15.35393526281676343624906856534, −14.55264596905564444810926365027, −13.53568085621500278981434639373, −12.702855168502902948144140435511, −11.79954746593644769943298823716, −11.2911630450741818366195336922, −10.29429542443639090340841989119, −9.28820554508382275905400396144, −8.925309873663989874149910830324, −8.10969509896377507210651053545, −7.52453396965288705471607431463, −6.521142892062161061445082731234, −5.75247482616960441742817392823, −4.8316056821805446982823286899, −4.33864970840809104306394040433, −2.742482122377030894856725168584, −1.86673429567303823328826921674, −0.89849205335874190834574074904,
0.19599849721323678655881936238, 0.99357853891680351432101316182, 2.09753016921106847085823067028, 3.18681004357566190846454354814, 3.70490982866820854093170115455, 4.43302380670983100708714772803, 6.00332816882446952779682496962, 6.706089407339202071626529184140, 7.629617608839571730004335591864, 8.219537130369355693098014732209, 8.73253251584485698470698647612, 10.15242126438922527555473852971, 10.56586285620383159340387588840, 11.02725389463555629787699380039, 11.734081218099949134302124767174, 12.80018076698676435194984784513, 13.37416044080808401756008799392, 14.259032713131434913679332384925, 15.171622192365612965963536668870, 15.9925077229445937843177095886, 16.59890444820667029301409300084, 17.41913099638473485664233105909, 18.110455508362449714972581549249, 18.75755085104707257798278649730, 19.519710930653637547479749594164