L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s + 11-s + i·12-s − i·13-s + 16-s + i·17-s + i·18-s + 19-s − i·22-s + i·23-s + 24-s + ⋯ |
L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s + 11-s + i·12-s − i·13-s + 16-s + i·17-s + i·18-s + 19-s − i·22-s + i·23-s + 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3495409364 - 0.6269381942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3495409364 - 0.6269381942i\) |
\(L(1)\) |
\(\approx\) |
\(0.6429206321 - 0.6213619628i\) |
\(L(1)\) |
\(\approx\) |
\(0.6429206321 - 0.6213619628i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + iT \) |
| 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
−36.219896895435381643065199808892, −34.96311678411762509296875963934, −33.76268559702125704427475143284, −32.9838331568376425808078271417, −31.93405194990359413445943294049, −30.84274987882135682248030961034, −28.74320900269888855242547135730, −27.4602651989291724224164615184, −26.65109622148002811558886546996, −25.48973812595570355198839599415, −24.22286625632355759831920275751, −22.690978629151542201098086262251, −21.9182490332402576380203841768, −20.31396637660893572313074337635, −18.57036781802642855189416732928, −16.9594712992140305343597984306, −16.196602777548024283312635350635, −14.817159354460680897899718186, −13.860554686079885706438368123375, −11.68350960227318976088753411709, −9.748062668948478834491865400376, −8.78447499880181833223265346231, −6.862885020203486068328139135464, −5.20395516969527222914434958646, −3.82563061124272229542433742487,
1.527642942546248751768372213376, 3.41948173395721615043337197672, 5.69811009151214174983702061306, 7.74177751885047395888392449373, 9.2554330327191394876459260761, 11.053478916741687481603872758834, 12.262193970198583798876640901354, 13.33076328509740566163682877234, 14.6330808216027073292113375627, 17.15671754869575145751683826601, 18.15074251271976619161510547886, 19.445296194174099175945502728693, 20.24072717160410528073128794936, 21.98639976815761551022228892823, 23.041337673810567924880400625073, 24.37413874887608792004563219858, 25.788321567429246981409529946750, 27.39878541779762140002021896241, 28.51995644866412899747302258081, 29.75398098699249022140680697189, 30.409235903601587578397042002710, 31.58000120611266879858115315066, 32.89805787003820697711851844002, 34.96594794508327945164366480788, 35.59951234669536118631027473796