L(s) = 1 | + (−0.5 − 1.65i)3-s + 3.31i·5-s + (−2.5 + 1.65i)9-s − 3.31i·11-s + (5.5 − 1.65i)15-s − 3.31i·23-s − 6·25-s + (4 + 3.31i)27-s + 5·31-s + (−5.5 + 1.65i)33-s + 7·37-s + (−5.5 − 8.29i)45-s + 6.63i·47-s + 7·49-s + 13.2i·53-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.957i)3-s + 1.48i·5-s + (−0.833 + 0.552i)9-s − 1.00i·11-s + (1.42 − 0.428i)15-s − 0.691i·23-s − 1.20·25-s + (0.769 + 0.638i)27-s + 0.898·31-s + (−0.957 + 0.288i)33-s + 1.15·37-s + (−0.819 − 1.23i)45-s + 0.967i·47-s + 49-s + 1.82i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.416826841\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.416826841\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 1.65i)T \) |
| 11 | \( 1 + 3.31iT \) |
good | 5 | \( 1 - 3.31iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.995902893883704581538423582107, −8.125761136656858807096737388411, −7.52157178913655488708262273592, −6.67187674572063256922838387360, −6.23746527664470394576221388969, −5.48452190932015925250047790558, −4.12654015985199926967680666551, −2.92271848694942299168589100823, −2.49283446278566176967077572663, −0.943791306607181523366046383854,
0.67769763463998855278241716335, 2.06362696760926683338813045412, 3.50644342887959080509545594916, 4.41287837147238924856156267408, 4.92137647007288042105875274189, 5.60130490387487089228959100742, 6.56945284981696285834975171300, 7.71470562543099578229914574172, 8.482599881198749229996234759299, 9.163606540702987546512431750872