L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s − 6-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (0.939 + 0.342i)13-s + (0.5 + 0.866i)14-s + (−0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s − 6-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (0.939 + 0.342i)13-s + (0.5 + 0.866i)14-s + (−0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2728970460 + 0.6091696721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2728970460 + 0.6091696721i\) |
\(L(1)\) |
\(\approx\) |
\(0.5730256230 + 0.6028616835i\) |
\(L(1)\) |
\(\approx\) |
\(0.5730256230 + 0.6028616835i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 + 0.342i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.939 + 0.342i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−35.3246143635791424308720588999, −34.598115521742330046183640873689, −32.17589015975588054808924110867, −31.22779458708764400247871823403, −30.500940037908529412695328711730, −29.18124676862525714569439556429, −28.12653650299418180341509909885, −27.11102290279824701784820907542, −25.41089174075348964331022724642, −23.99340056835739397546274868979, −23.095673111181431982202952528895, −21.22532623007380492916163247380, −20.2857075846320586014774606501, −18.90854624061228455909847907802, −18.294280900585872254205705913206, −16.63398122301660007436649276387, −14.5248929595323202146958025779, −13.02738078356333233080063428918, −12.09429545145793591479221093531, −10.97700886914988791975009604334, −8.58139326078342803618355406973, −8.10272357707850378095747145648, −5.420362327351430206624935192594, −3.32290403676842290931058032293, −1.36111872145182545173201029804,
3.79770706941784267078472872641, 5.05529475560601806138394609841, 7.15317776955223512678826869163, 8.35020742300255258348190221368, 9.99471994821601930946177458734, 11.24575579035053140899260201945, 13.76162252981835000942418011962, 14.92992840018631673601381585473, 15.7187005940795798310575442379, 17.05280211805387660463422622819, 18.39327756383791419107886546660, 19.97289309798799357463800245099, 21.43418170723832982566322134725, 23.02220995021267017846663693034, 23.557185588307763619515639743372, 25.52412406137941537805807082375, 26.37811385004794953255824231267, 27.35234622302266576426852016860, 28.1568692134841556434653762626, 30.58374534777128762718162974694, 31.42460828509511208582358851621, 32.8104936441167454980384843732, 33.72370093807847105049095112730, 34.450878824919384579074961041155, 35.98481981469406901194170491145