L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.951 − 0.309i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)6-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.913 − 0.406i)11-s + (0.743 + 0.669i)12-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (0.406 + 0.913i)17-s + (−0.743 + 0.669i)18-s + (0.978 + 0.207i)19-s + (0.978 + 0.207i)21-s + (0.587 − 0.809i)22-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.951 − 0.309i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)6-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.913 − 0.406i)11-s + (0.743 + 0.669i)12-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (0.406 + 0.913i)17-s + (−0.743 + 0.669i)18-s + (0.978 + 0.207i)19-s + (0.978 + 0.207i)21-s + (0.587 − 0.809i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4765019922 + 0.1632270890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4765019922 + 0.1632270890i\) |
\(L(1)\) |
\(\approx\) |
\(0.4968549657 + 0.1740329804i\) |
\(L(1)\) |
\(\approx\) |
\(0.4968549657 + 0.1740329804i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 7 | \( 1 + (-0.994 + 0.104i)T \) |
| 11 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.406 + 0.913i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (-0.406 - 0.913i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (-0.207 + 0.978i)T \) |
| 47 | \( 1 + (-0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.406 - 0.913i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.994 + 0.104i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (-0.207 + 0.978i)T \) |
| 89 | \( 1 + (-0.104 - 0.994i)T \) |
| 97 | \( 1 + (0.406 - 0.913i)T \) |
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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
−20.01580277176309366832053695223, −19.01750789788908947015790913592, −18.40117321965867458948644618611, −17.84886489689695361135176847829, −17.11099052688283955882617811267, −16.13031454710031348874963486481, −15.92947517178150108547057473975, −14.73127615533396685466290976915, −13.57673807090555421318381944029, −13.102634480110646549797316172143, −12.29777526788842895470460438462, −11.747002097354066375821745395006, −10.93355337749021634793722063207, −10.25471155483807716011336459349, −9.557813315851955466647772592838, −9.20742738984891726156495747176, −7.64670927291108458655917485443, −7.24365006816564515268972752635, −5.89271324310134121052302920871, −5.294660615324987781318977251312, −4.46797270346092903100518182228, −3.50590273895532336801274850613, −2.86479265623942538010889424074, −1.6409942839646239867769947403, −0.51704285107465739570131928068,
0.45050856518595484145525415991, 1.616840136355241208981591597574, 3.06917709537190799360916373409, 4.0897437277330288046525985172, 5.06726993629718716631265146019, 5.821970599594518416601139372, 6.22261109780246325392496537684, 7.1179233488292151987739356321, 7.81863239792736727605432184360, 8.57593591357580339180614165767, 9.77427269905987184546775290535, 10.12987062859526509860815458767, 10.9963603431286706336270285833, 12.05453654723683004566187552274, 12.95361697145375219483848576813, 13.18727113414054569738974757776, 14.196948946609956300172872526227, 15.14886631033445640159637112910, 15.94257170306656021573496285031, 16.36559918472653242019222491750, 16.9199171856454529843449021952, 17.818550270873785490512946035775, 18.45730110471401756619635056566, 18.93801046051180449668237512650, 19.64267658955376066776077934278