L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 18-s − 19-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 18-s − 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1022807544 - 0.08993978631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1022807544 - 0.08993978631i\) |
\(L(1)\) |
\(\approx\) |
\(0.3465398303 - 0.3578308262i\) |
\(L(1)\) |
\(\approx\) |
\(0.3465398303 - 0.3578308262i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 3 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + (0.900 + 0.433i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.900 - 0.433i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (-0.900 + 0.433i)T \) |
| 59 | \( 1 + (-0.623 + 0.781i)T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.222 - 0.974i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.222 - 0.974i)T \) |
| 97 | \( 1 - T \) |
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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
−34.32132191791602344623166785396, −33.54442619454019371686546168076, −32.419662208635052530849446075536, −31.36048576416160121411052521903, −29.85405192250389225320542923475, −27.98815417174672432578005475061, −27.58172383647540370428405068348, −26.323572192553112713906426806820, −25.47682069789491921971331773244, −23.64666608843425817131494988756, −22.932980441360385057175933783970, −22.07045335029736531304572323858, −20.29903368798745191200525557741, −18.59648099722324866772376323050, −17.64854010516645228562248000575, −16.34951559465512605939900101800, −15.29601709252427400113956889433, −14.577539852530280407368602386910, −12.5062837505297389763323536249, −10.76048714437120514921038290753, −9.79019251964555466950469929460, −8.00884983912041497385627311933, −6.6511913232285671756233503405, −5.2235870049862019131840397596, −3.767150760966424750135916042970,
0.09468210403770850591718334042, 1.68366398596360998847434856293, 3.9255330430267476667620634680, 5.598758255829638354504644268362, 7.73790205128176591311583814908, 8.871511114139271950507747319070, 10.79358360510946526622781435530, 11.824299840207162009259677376349, 12.743774063369691765024512201331, 13.91567116801008849611805413120, 16.34025813678441105045203585633, 17.21969445983074282213740940504, 18.813445915931509027239848592783, 19.30796500154226098349333376649, 20.810956834655686726207779722600, 21.97489319626256060309273023807, 23.458746317477307567237136768192, 24.04831935121641523476395696938, 25.85121991079588928671589939455, 27.42159671426839651607334626352, 28.21032138635890293142640010271, 29.18972083112871472095794891490, 30.15188589711422934762287854353, 31.31821818457551084958004083870, 32.21869067791450474824643325273