L(s) = 1 | + i·3-s − i·7-s − 9-s + 2·13-s + 2i·19-s + 21-s + 25-s − i·27-s + 2i·39-s − 49-s − 2·57-s − 2·61-s + i·63-s + i·75-s − 2i·79-s + ⋯ |
L(s) = 1 | + i·3-s − i·7-s − 9-s + 2·13-s + 2i·19-s + 21-s + 25-s − i·27-s + 2i·39-s − 49-s − 2·57-s − 2·61-s + i·63-s + i·75-s − 2i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128234316\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128234316\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 2T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.11200839042751556043772798304, −9.102455760899311584670151755378, −8.402172298659248132592523459432, −7.67004482517405927059557654228, −6.38460940670539142674041580352, −5.82341410171504914367181872037, −4.65190651024797205426139835890, −3.80971667961244198549523359263, −3.28439258559214431625703184604, −1.40486403518532242249004867753,
1.20403815503451087888988435018, 2.47495016111105270558228759483, 3.32011807519595132741865495142, 4.79153249099958305568443874700, 5.77339543976341204223818388542, 6.42183063986854377059349482217, 7.13977745847292536294915545202, 8.292130196846028609239823856448, 8.741660049457947189601230705677, 9.354954356029106958045375089875