L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s − i·7-s + i·8-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + i·20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s − i·7-s + i·8-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + i·20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6097151304 - 0.09541121980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6097151304 - 0.09541121980i\) |
\(L(1)\) |
\(\approx\) |
\(0.6659085684 + 0.01812298310i\) |
\(L(1)\) |
\(\approx\) |
\(0.6659085684 + 0.01812298310i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + iT \) |
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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
−28.77707118021407782082172749194, −28.31297033443049086022366364991, −27.35082337663416254877980143746, −26.55675902683610324546969516642, −25.09445860542703143753469297933, −24.64239931826080157680762831319, −23.03046535865268135772117541865, −21.90110438956894177758770414299, −20.85257661148289001444012767309, −19.75901200570919929988524303925, −19.13956087671786475707089229210, −17.9654700775053432098369799196, −16.90455498071493264772862073886, −15.83480974093782338799396895476, −14.932677948555566979070382611478, −12.91112399201234958335455337528, −12.02280150262734112948390054003, −11.31125637712081166911432239008, −9.7219415762604806025881014652, −8.767027023739913690293585000904, −7.86232130661416179267335564940, −6.441162056626149070507594049913, −4.52350064755463807081016286829, −3.126606160340175363150960216070, −1.470836900818648603480513859108,
0.90360310413768660320602497964, 3.13200009455603255490952542681, 4.73856718728446611510605401659, 6.62369966009090637940237394090, 7.26280442863311733573989314731, 8.46804424371464918387631553676, 9.71341914246516039773153384336, 10.98436100907993124206315741654, 11.63677570167886252333917460778, 13.683575383397319227491988424887, 14.649656629177603400488570818840, 15.79214957445637346570972383305, 16.649529503472134532078457205972, 17.71370460576671670601690332884, 18.84492815114390041860636665850, 19.734010290974155775426237408923, 20.448313216120179497229098019875, 22.30515441311615030657687307415, 23.273975153403488858832193914277, 24.157084695354564311263670718, 25.16959362446638525517574857942, 26.56234234769849124719504825321, 26.876421609609381191125651749364, 27.783698698199183247804266442010, 29.11783234602121659237004790661