L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.286 − 0.957i)3-s + (0.173 − 0.984i)4-s + (0.998 − 0.0581i)5-s + (0.396 + 0.918i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (−0.893 − 0.448i)13-s + (−0.973 + 0.230i)14-s + (0.230 − 0.973i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.286 − 0.957i)3-s + (0.173 − 0.984i)4-s + (0.998 − 0.0581i)5-s + (0.396 + 0.918i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (−0.893 − 0.448i)13-s + (−0.973 + 0.230i)14-s + (0.230 − 0.973i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.850908980 - 0.1324549518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.850908980 - 0.1324549518i\) |
\(L(1)\) |
\(\approx\) |
\(1.097362239 + 0.02623520983i\) |
\(L(1)\) |
\(\approx\) |
\(1.097362239 + 0.02623520983i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.286 - 0.957i)T \) |
| 5 | \( 1 + (0.998 - 0.0581i)T \) |
| 7 | \( 1 + (0.893 + 0.448i)T \) |
| 11 | \( 1 + (0.727 + 0.686i)T \) |
| 13 | \( 1 + (-0.893 - 0.448i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.957 - 0.286i)T \) |
| 31 | \( 1 + (0.549 - 0.835i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.342 + 0.939i)T \) |
| 47 | \( 1 + (0.802 - 0.597i)T \) |
| 53 | \( 1 + (0.549 + 0.835i)T \) |
| 59 | \( 1 + (-0.973 - 0.230i)T \) |
| 61 | \( 1 + (0.116 - 0.993i)T \) |
| 67 | \( 1 + (0.998 + 0.0581i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.686 - 0.727i)T \) |
| 79 | \( 1 + (-0.893 + 0.448i)T \) |
| 83 | \( 1 + (0.286 - 0.957i)T \) |
| 89 | \( 1 + (-0.116 + 0.993i)T \) |
| 97 | \( 1 + (0.998 - 0.0581i)T \) |
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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
−18.49991590271893890755138833602, −17.586616805779421589309399056646, −17.15907916231552852495064501690, −16.79509273492429074097554681242, −15.96945399082949260938931782382, −15.06064589701158923592770237648, −14.25445176152948803605314800997, −13.86251827599179203593080703138, −13.087363685963950771813571101752, −12.014782686844096776696756583289, −11.32727257855756218249680989942, −10.774375615476809083655906838449, −10.23391517734389418699377712322, −9.46357797860668087800777314015, −8.85545521451783054140090421789, −8.54372039593554065818200906287, −7.33805616750068456160417000886, −6.79292259320480368386382757063, −5.59562308642765096091975506899, −4.81488595803987564919073951402, −4.1484767328301519121481984987, −3.283591423874436615945997356520, −2.45579432304696112737810150723, −1.81438754242428333848270464227, −0.82768247248852148985024170649,
0.8449656269319145602837220462, 1.66294160643535839100209343222, 2.1258640799738995135296733769, 2.92331892978991484135458273270, 4.57866621723552135958823144072, 5.26069595511342129432049294016, 5.92084942284549286235830321265, 6.637633872479203347675843143034, 7.38679314867586128004179090167, 7.82750740152301203334600302879, 8.73214434296415099104682932722, 9.44136179856298781021277535835, 9.66826268063297881391659851940, 10.89254154333178659321061688142, 11.567793629359793685987056944611, 12.22281305503612592065814678144, 13.243395352393146453241885990756, 13.73875871247742006960279976674, 14.601778788148100706203223319813, 14.8519095420472347875844825892, 15.605200225033549178489828750894, 17.00181520273881024671777814078, 17.12769209419943626234441359413, 17.77589509159183918072576368140, 18.321441515361917822285450466878