L(s) = 1 | + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.623 − 0.781i)6-s + (0.974 + 0.222i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.433 − 0.900i)11-s − 12-s + (0.433 + 0.900i)13-s + (0.781 − 0.623i)14-s + (−0.900 + 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (−0.974 + 0.222i)19-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.623 − 0.781i)6-s + (0.974 + 0.222i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.433 − 0.900i)11-s − 12-s + (0.433 + 0.900i)13-s + (0.781 − 0.623i)14-s + (−0.900 + 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (−0.974 + 0.222i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6305830952 - 1.442252126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6305830952 - 1.442252126i\) |
\(L(1)\) |
\(\approx\) |
\(1.028023846 - 1.052911572i\) |
\(L(1)\) |
\(\approx\) |
\(1.028023846 - 1.052911572i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 3 | \( 1 + (0.222 - 0.974i)T \) |
| 7 | \( 1 + (0.974 + 0.222i)T \) |
| 11 | \( 1 + (-0.433 - 0.900i)T \) |
| 13 | \( 1 + (0.433 + 0.900i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.974 + 0.222i)T \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 31 | \( 1 + (0.781 + 0.623i)T \) |
| 37 | \( 1 + (0.900 + 0.433i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (-0.623 - 0.781i)T \) |
| 47 | \( 1 + (0.900 - 0.433i)T \) |
| 53 | \( 1 + (0.781 + 0.623i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.974 - 0.222i)T \) |
| 67 | \( 1 + (0.433 - 0.900i)T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.433 + 0.900i)T \) |
| 83 | \( 1 + (0.974 - 0.222i)T \) |
| 89 | \( 1 + (-0.781 - 0.623i)T \) |
| 97 | \( 1 + (0.222 + 0.974i)T \) |
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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
−28.14088596714089517862805088785, −27.53402603688303830771850066639, −26.447691855034622977305481140594, −25.655575443305390135284151680, −24.79972790359526090078111788278, −23.45223082161391286439571359794, −22.863821783013171098605148080124, −21.63413193302341786778900427976, −20.89400314970013836431348938167, −20.15716300728641340959742818520, −18.19761166580419769248196991988, −17.28003598384811120465571353397, −16.33653947311142403421823469875, −15.16071005703034577076583098712, −14.747054007163167241789154971069, −13.599339110338391031685016134124, −12.3402593762093414941217285692, −11.02529051683120778686645967006, −9.88080956100224071295635170335, −8.36123549643266075416058353564, −7.748629658781018197385039695093, −5.9869000046482701838563763818, −4.881450691793828598730375018801, −4.08462951205853870020614934832, −2.620694742378001164200746867245,
1.29499788792136773980683591853, 2.38902129092622124499046056091, 3.793290583127576453943588219907, 5.36439776887953451267871564604, 6.34952802021737144556405088901, 7.96727746240044326339263787973, 8.951922116215014407019728668410, 10.6123689654806258928769297978, 11.63160500865977751084379102821, 12.34191265317921808840184852734, 13.71633996382606264931202089796, 14.10880824718808789207578475115, 15.3225117746274136016798506559, 16.998784720915251779033219803947, 18.416813756042589012883622433765, 18.79366190063372639399004818214, 19.91508816930276148543707129638, 21.0690178017438960951784646032, 21.615843910030804445690908476073, 23.26014428481929504927715661137, 23.76101476508410915740697601825, 24.56330549125811741522618888902, 25.73821721124752099528712646594, 27.131663979625472704188658974368, 28.17247982518285245873559792112