L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.07997307365 + 0.2125045006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.07997307365 + 0.2125045006i\) |
\(L(1)\) |
\(\approx\) |
\(0.6801025146 + 0.3158538266i\) |
\(L(1)\) |
\(\approx\) |
\(0.6801025146 + 0.3158538266i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.978 - 0.207i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (-0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.104 - 0.994i)T \) |
| 53 | \( 1 + (-0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.104 - 0.994i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−21.38618628383407791198458822810, −20.46393930128202591370869542199, −19.72940133656722163691730721215, −18.929501643219726149815935250548, −18.23305298596389877284176239096, −17.28963097953501070220442404999, −16.20922606509003429178806148129, −15.61893676049288945217233945882, −14.87653365530905741507775748129, −14.109612033169101631607275639046, −12.76232664448683055756943807892, −12.40063922225604028724414505813, −11.47954845403517311792334612302, −10.9793761291737776983044060925, −10.23850544947037101409235996286, −9.36139232714986437106768801417, −8.1786976296212733423472141547, −6.93209135247560491710081379636, −6.20230027309809150608732146755, −5.198829473738535195108116622403, −4.40577668129908669767918116712, −3.74473328074075156950947191686, −2.77337549402075497864178512688, −1.30485312030714227535857713848, −0.09462079218329048534022657985,
1.51451634888068864830586147280, 3.14205204892777059691960700889, 3.98409824853628315627935059387, 4.97775099492757383247201544241, 5.5535000647526669059505161934, 6.57495065350702186528902846604, 7.42150222017451910290039740616, 7.87400340145482206083782519710, 8.93133525808398130499037113641, 10.25095095220017615647841649396, 11.311453922458245888254978148541, 12.09426670749836553826911976246, 12.3486262356732962692852695205, 13.41149671091699408538805598989, 14.21126192568970924837006715790, 15.16772992164325252291210757371, 16.10670435860512789208327912311, 16.3394696073478338934736615424, 17.235307986516258398589053642327, 18.12935144362035234895684430947, 18.77174135709833541627521991586, 19.81518768262104229468142571805, 20.74994525611837979011003499502, 21.63076970616632810905479592260, 22.42310285655633378674827205368