L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + i·5-s + (0.707 + 0.707i)6-s − i·8-s − i·9-s − 10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (0.707 + 0.707i)15-s + 16-s + (0.707 + 0.707i)17-s + 18-s + (0.707 + 0.707i)19-s + ⋯ |
L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + i·5-s + (0.707 + 0.707i)6-s − i·8-s − i·9-s − 10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (0.707 + 0.707i)15-s + 16-s + (0.707 + 0.707i)17-s + 18-s + (0.707 + 0.707i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.368209078 + 0.6279209252i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368209078 + 0.6279209252i\) |
\(L(1)\) |
\(\approx\) |
\(1.198765398 + 0.4259710896i\) |
\(L(1)\) |
\(\approx\) |
\(1.198765398 + 0.4259710896i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (0.707 + 0.707i)T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.707 + 0.707i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.707 - 0.707i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + (-0.707 - 0.707i)T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 + (-0.707 - 0.707i)T \) |
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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
−25.570103984699882836156693490687, −24.57967081459016728706368754970, −23.42162905654474626486293478175, −22.39792017134838650041732487990, −21.53293895438128691234091269103, −20.68088404738941600014927310696, −20.21670665470094247202424895940, −19.418575287051530689743410320574, −18.34960259447904489524213685954, −17.13530674362817917620065884385, −16.29147110437768598575602475732, −15.183730632104248449508130106861, −13.92379042769314092578856362355, −13.50336892567475175594554368699, −12.14705430181518892448671868395, −11.488506997667179043614416525, −10.06837111154444168822299881871, −9.39213511834289108078667350317, −8.74242469617326919347508608342, −7.63122166293401536204014827081, −5.57912070648586491329381971176, −4.45435109218744651477375993809, −3.89337955593729565446729635976, −2.47983555450524682076211209796, −1.291894744594250163221301046834,
1.28998706782066233540883555329, 3.15935316400359031242977899972, 3.83897053458605363938836306821, 5.91928401062067203834244718023, 6.32142673380223484172344420203, 7.63160063013837353966487598230, 8.11729391437334818263999291682, 9.3045389135751190141477176730, 10.36001766437339342403738576206, 11.82301307835439125544355833493, 12.98831583079696038090321843443, 13.97223648336331208858648910724, 14.45314508601531803106108438486, 15.32069971749511352249371324263, 16.392757484532219408745846027299, 17.606066637685191739882699428905, 18.37399890821025682464240903194, 18.96415497554738776860034641550, 19.932303994956969624615138429872, 21.310431467326596676994583630476, 22.33744666026454048038321786468, 23.1310833437089750076737661838, 23.999143568929462936662372905888, 24.88806028666684403094000967781, 25.59188618307252832312743475945