L(s) = 1 | + (0.937 − 0.347i)2-s + (−0.918 − 0.394i)3-s + (0.758 − 0.651i)4-s + (−0.968 + 0.250i)5-s + (−0.998 − 0.0506i)6-s + (0.440 − 0.897i)7-s + (0.485 − 0.874i)8-s + (0.688 + 0.724i)9-s + (−0.820 + 0.571i)10-s + (0.485 + 0.874i)11-s + (−0.954 + 0.299i)12-s + (0.874 − 0.485i)13-s + (0.101 − 0.994i)14-s + (0.988 + 0.151i)15-s + (0.151 − 0.988i)16-s + (0.758 − 0.651i)17-s + ⋯ |
L(s) = 1 | + (0.937 − 0.347i)2-s + (−0.918 − 0.394i)3-s + (0.758 − 0.651i)4-s + (−0.968 + 0.250i)5-s + (−0.998 − 0.0506i)6-s + (0.440 − 0.897i)7-s + (0.485 − 0.874i)8-s + (0.688 + 0.724i)9-s + (−0.820 + 0.571i)10-s + (0.485 + 0.874i)11-s + (−0.954 + 0.299i)12-s + (0.874 − 0.485i)13-s + (0.101 − 0.994i)14-s + (0.988 + 0.151i)15-s + (0.151 − 0.988i)16-s + (0.758 − 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.209136567 - 2.086643639i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.209136567 - 2.086643639i\) |
\(L(1)\) |
\(\approx\) |
\(1.229615465 - 0.6962578320i\) |
\(L(1)\) |
\(\approx\) |
\(1.229615465 - 0.6962578320i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 373 | \( 1 \) |
good | 2 | \( 1 + (0.937 - 0.347i)T \) |
| 3 | \( 1 + (-0.918 - 0.394i)T \) |
| 5 | \( 1 + (-0.968 + 0.250i)T \) |
| 7 | \( 1 + (0.440 - 0.897i)T \) |
| 11 | \( 1 + (0.485 + 0.874i)T \) |
| 13 | \( 1 + (0.874 - 0.485i)T \) |
| 17 | \( 1 + (0.758 - 0.651i)T \) |
| 19 | \( 1 + (-0.988 + 0.151i)T \) |
| 23 | \( 1 + (0.394 + 0.918i)T \) |
| 29 | \( 1 + (0.979 - 0.201i)T \) |
| 31 | \( 1 + (0.954 + 0.299i)T \) |
| 37 | \( 1 + (-0.528 - 0.848i)T \) |
| 41 | \( 1 + (-0.0506 + 0.998i)T \) |
| 43 | \( 1 + (-0.651 - 0.758i)T \) |
| 47 | \( 1 + (-0.101 - 0.994i)T \) |
| 53 | \( 1 + (-0.724 - 0.688i)T \) |
| 59 | \( 1 + (-0.979 + 0.201i)T \) |
| 61 | \( 1 + (0.394 - 0.918i)T \) |
| 67 | \( 1 + (-0.968 + 0.250i)T \) |
| 71 | \( 1 + (0.758 + 0.651i)T \) |
| 73 | \( 1 + (0.688 - 0.724i)T \) |
| 79 | \( 1 + (-0.571 - 0.820i)T \) |
| 83 | \( 1 + (0.688 - 0.724i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.651 + 0.758i)T \) |
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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
−24.22236373523341501561916103074, −23.82016718232758666118815920838, −22.9782979475817193347783781349, −22.20375976571845887810573144021, −21.233791356945159359014915032677, −20.92391674289863858470283826147, −19.37643053685013061044302468052, −18.57489212464062237536396748615, −17.17365252769058096332141690116, −16.54699785567945296519427932317, −15.701010161958319136032113513566, −15.11740919140247334318069275838, −14.11647875203922362494751677345, −12.70375820222034131389426413168, −12.090386785153783588521223404999, −11.35477247563920197963370367724, −10.681473347739930030591045752173, −8.777852118907862778830047680204, −8.1498760490265863051699192834, −6.59077326654284132540145100134, −6.03291161394921715668697759404, −4.868132017773953019003562843559, −4.157231978934825960345965766060, −3.11158418954185160752169121903, −1.249671747833233332737252068435,
0.63344358560365568575998253263, 1.64288776596678135507192547640, 3.36200243079582126050196758738, 4.32669273799150849383004214667, 5.0774812694624387938380807011, 6.43803957966909342983066846287, 7.127252049731025569587838762349, 8.01741176901138415185768840857, 10.08048745850953508498442627029, 10.7838528919559308574760925009, 11.60124620777556482882295683917, 12.22601332695943763490544621292, 13.19423718108887969726509575004, 14.129051544690738418075580755439, 15.17139720497877960335004508381, 15.97144024099796223522982302695, 16.9334602063477594274201425592, 17.943830978515808641596792963278, 19.03651776208219661375181216007, 19.77438923317638849756413315915, 20.685322044146213640688636324117, 21.581830959425595106795903353158, 22.86839777772380251282180254496, 23.17995897578673733069071459661, 23.51711270648340255473302078718