L(s) = 1 | + (−0.647 − 0.762i)2-s + (0.796 − 0.605i)3-s + (−0.161 + 0.986i)4-s + (0.468 − 0.883i)5-s + (−0.976 − 0.214i)6-s + (−0.994 + 0.108i)7-s + (0.856 − 0.515i)8-s + (0.267 − 0.963i)9-s + (−0.976 + 0.214i)10-s + (0.947 − 0.319i)11-s + (0.468 + 0.883i)12-s + (−0.267 − 0.963i)13-s + (0.725 + 0.687i)14-s + (−0.161 − 0.986i)15-s + (−0.947 − 0.319i)16-s + (−0.994 − 0.108i)17-s + ⋯ |
L(s) = 1 | + (−0.647 − 0.762i)2-s + (0.796 − 0.605i)3-s + (−0.161 + 0.986i)4-s + (0.468 − 0.883i)5-s + (−0.976 − 0.214i)6-s + (−0.994 + 0.108i)7-s + (0.856 − 0.515i)8-s + (0.267 − 0.963i)9-s + (−0.976 + 0.214i)10-s + (0.947 − 0.319i)11-s + (0.468 + 0.883i)12-s + (−0.267 − 0.963i)13-s + (0.725 + 0.687i)14-s + (−0.161 − 0.986i)15-s + (−0.947 − 0.319i)16-s + (−0.994 − 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3268725709 - 1.249374812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3268725709 - 1.249374812i\) |
\(L(1)\) |
\(\approx\) |
\(0.7060556766 - 0.6883159728i\) |
\(L(1)\) |
\(\approx\) |
\(0.7060556766 - 0.6883159728i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.647 - 0.762i)T \) |
| 3 | \( 1 + (0.796 - 0.605i)T \) |
| 5 | \( 1 + (0.468 - 0.883i)T \) |
| 7 | \( 1 + (-0.994 + 0.108i)T \) |
| 11 | \( 1 + (0.947 - 0.319i)T \) |
| 13 | \( 1 + (-0.267 - 0.963i)T \) |
| 17 | \( 1 + (-0.994 - 0.108i)T \) |
| 19 | \( 1 + (0.0541 + 0.998i)T \) |
| 23 | \( 1 + (-0.907 - 0.419i)T \) |
| 29 | \( 1 + (0.647 - 0.762i)T \) |
| 31 | \( 1 + (-0.0541 + 0.998i)T \) |
| 37 | \( 1 + (0.856 + 0.515i)T \) |
| 41 | \( 1 + (0.907 - 0.419i)T \) |
| 43 | \( 1 + (0.947 + 0.319i)T \) |
| 47 | \( 1 + (-0.468 - 0.883i)T \) |
| 53 | \( 1 + (0.976 + 0.214i)T \) |
| 61 | \( 1 + (-0.647 - 0.762i)T \) |
| 67 | \( 1 + (0.856 - 0.515i)T \) |
| 71 | \( 1 + (0.468 + 0.883i)T \) |
| 73 | \( 1 + (0.725 + 0.687i)T \) |
| 79 | \( 1 + (0.796 + 0.605i)T \) |
| 83 | \( 1 + (0.370 - 0.928i)T \) |
| 89 | \( 1 + (-0.647 + 0.762i)T \) |
| 97 | \( 1 + (0.725 - 0.687i)T \) |
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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
−33.04218391567308082284788620790, −32.20270844375283533649991731756, −30.87563025490508879000525737949, −29.436886679887358012769345320413, −28.19473130232338978463606327318, −26.89339573313098085568869017467, −26.09224802274564694343013574849, −25.5493492141813174035420592770, −24.27752158375275963582348639233, −22.58239631117645014234006186405, −21.80799708940594845125590855318, −19.832736261182619546868642400280, −19.308472592664960208470769383040, −17.88088823291300872993359063747, −16.565387489952513167615350629500, −15.47387850834109091556969204390, −14.43348497742345429469141421726, −13.5349399632591434880641896531, −10.99744940194300360711660271778, −9.66594229344762944424516043773, −9.14873209319883639395860592499, −7.27029959005301961429188816508, −6.29415089612904760546828306433, −4.21367241701027987850101752143, −2.30128744550150201276043219007,
0.802669032444658459818351810168, 2.38686363680666098786827855660, 3.87799384118786592699244967022, 6.38838178094025250015876767485, 8.09785093982188965076072609516, 9.094029344105642121786949760770, 10.02236701390910212026484687854, 12.12140801382567929941969315357, 12.85628187031796143528106235484, 13.92020060312200074582736996275, 15.93917997256474580089091996054, 17.218945212099607603551710515536, 18.33352285307464604107566707534, 19.71649057771209881465420657780, 20.05471121792166551249108189263, 21.408073816134084433073713806658, 22.66925998111074756169912494689, 24.67378497923962921305224946228, 25.211299661340871523104683174532, 26.387279497214778258780638389329, 27.58546963313497261454827755204, 28.957413397731240590797622527834, 29.52468787387764997382153463731, 30.67603383874288628214144042879, 31.934542621238848419191046159062