L(s) = 1 | + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s − 17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s − 17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6356265095 + 0.01194645729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6356265095 + 0.01194645729i\) |
\(L(1)\) |
\(\approx\) |
\(0.5823855984 + 0.3076617941i\) |
\(L(1)\) |
\(\approx\) |
\(0.5823855984 + 0.3076617941i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−30.11631440110346509418331112471, −29.29812849659782908447807133159, −28.20575006767634642763830669946, −27.30917411967729900919878301593, −26.29337233602426408537612002972, −24.58544011413190460732899556165, −23.623468105001233593304404399471, −22.55152135992932100010565547115, −22.013827128378420384386335223844, −20.173360507312202240452332335113, −19.459061153816189561985548119002, −18.533984855641599624457872521147, −17.66254423801616806717654411804, −16.27753909474791623988712570096, −14.48575420907922143436591159640, −13.49401765758018540819670923011, −12.12197343515599115733884613023, −11.560483784994570288345226309010, −10.51979942041702211398837199289, −8.77498357214516819284728767921, −7.5589975732424914055226760005, −6.095956927905613990310074469076, −4.35679243114058050134427680424, −2.91616828886730908262119832622, −1.224380782863076229195410390199,
0.36850103375058453678410952099, 3.896465100090453434409387593658, 4.62468018496037964137857283481, 5.998982900585529780379965115389, 7.37193079057648534394438006155, 8.79653683502309624879354980881, 9.67559707339583253429008005219, 11.35217199532715827395382786246, 12.46393510037638486996708551072, 14.0820442043343591148287113063, 15.368406867743075861953090186294, 15.87101467425224015785443480172, 16.99958554648530645137595190475, 17.81377593408914904337083661936, 19.459446109238196348971463134436, 20.59479548634250832078318185716, 22.16834070926495670839025572994, 22.70353508417886555331705208838, 23.87155244281167796661833392914, 24.7096373003145296566382468606, 26.19866010796777486811150698692, 26.93102778382724864215147912293, 27.91102929442649137793500921257, 28.49734977871951720587364110484, 30.46565195753053927729655771657