L(s) = 1 | + (−0.893 + 0.448i)5-s + (0.286 − 0.957i)7-s + (−0.0581 − 0.998i)11-s + (−0.597 − 0.802i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (0.286 + 0.957i)23-s + (0.597 − 0.802i)25-s + (−0.396 − 0.918i)29-s + (0.686 − 0.727i)31-s + (0.173 + 0.984i)35-s + (−0.173 + 0.984i)37-s + (−0.993 + 0.116i)41-s + (−0.835 + 0.549i)43-s + (0.686 + 0.727i)47-s + ⋯ |
L(s) = 1 | + (−0.893 + 0.448i)5-s + (0.286 − 0.957i)7-s + (−0.0581 − 0.998i)11-s + (−0.597 − 0.802i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (0.286 + 0.957i)23-s + (0.597 − 0.802i)25-s + (−0.396 − 0.918i)29-s + (0.686 − 0.727i)31-s + (0.173 + 0.984i)35-s + (−0.173 + 0.984i)37-s + (−0.993 + 0.116i)41-s + (−0.835 + 0.549i)43-s + (0.686 + 0.727i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09917152778 - 0.7260640362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09917152778 - 0.7260640362i\) |
\(L(1)\) |
\(\approx\) |
\(0.8261543087 - 0.1898860235i\) |
\(L(1)\) |
\(\approx\) |
\(0.8261543087 - 0.1898860235i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.893 + 0.448i)T \) |
| 7 | \( 1 + (0.286 - 0.957i)T \) |
| 11 | \( 1 + (-0.0581 - 0.998i)T \) |
| 13 | \( 1 + (-0.597 - 0.802i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.286 + 0.957i)T \) |
| 29 | \( 1 + (-0.396 - 0.918i)T \) |
| 31 | \( 1 + (0.686 - 0.727i)T \) |
| 37 | \( 1 + (-0.173 + 0.984i)T \) |
| 41 | \( 1 + (-0.993 + 0.116i)T \) |
| 43 | \( 1 + (-0.835 + 0.549i)T \) |
| 47 | \( 1 + (0.686 + 0.727i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.0581 + 0.998i)T \) |
| 61 | \( 1 + (-0.973 + 0.230i)T \) |
| 67 | \( 1 + (0.396 - 0.918i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.993 + 0.116i)T \) |
| 83 | \( 1 + (-0.993 - 0.116i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.893 + 0.448i)T \) |
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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
−23.077315981938520140226931237757, −22.30478447661496845728091430523, −21.33849249872863672474309894816, −20.51069133410282742646426792694, −19.88252894983964849240245134601, −18.731515131685494427880166840262, −18.42304163522052043408384653351, −17.1480624369549494018524666607, −16.36092593737387118755630495659, −15.58597505288562256793943486422, −14.78486268448601216923223157771, −14.07989653662028714762026441251, −12.5684603515081686326443939061, −12.17506256043496764937797230757, −11.555928379217482961430336158401, −10.27526149377615276773532811035, −9.29294870760584628014998841654, −8.55976310291473584819757183163, −7.55568704285729591196993089021, −6.8449037829966609680649585586, −5.31307792091901067188009819818, −4.82411531820305223133885162896, −3.66311276476646393045969398483, −2.48531833264975620070834244253, −1.34472075325502115044754626996,
0.19646702668997590391933359354, 1.16827199620833327698634903425, 2.94828866707502416608612870726, 3.56767209062078954808632640022, 4.63946831927812887801916769596, 5.71524659375090626341213509212, 6.89316540839760930911511885530, 7.78281490318867195103850041587, 8.20215557630451020447345616800, 9.6948748090497552348782683871, 10.50734818837556540753883989484, 11.33084335327986877686142049241, 11.96649659423467805442691886750, 13.2566816656713435449573632329, 13.8623588363502300698150968128, 14.95987407916843656977539177805, 15.48420470233418560326893944757, 16.61682237125563109137639011389, 17.2059316729451451544135892988, 18.24696888029955483739371458044, 19.183513540689908319708655831189, 19.719070017415553521951857866795, 20.55146501846588505848087342823, 21.50352121715624609626059959095, 22.43543934450383156880107366238