L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 + 0.866i)13-s + (−1.5 + 0.866i)19-s + 25-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (1.5 − 0.866i)91-s + 1.73i·103-s + (−0.5 + 0.866i)109-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 + 0.866i)13-s + (−1.5 + 0.866i)19-s + 25-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (1.5 − 0.866i)91-s + 1.73i·103-s + (−0.5 + 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.322829509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.322829509\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.979441483544332929766902880955, −8.442678181910493699750935129733, −7.75667065823235830401726484025, −6.67215960365943871774579036999, −6.31147507001122015262764068857, −5.14550878103349304957407066952, −4.13632102021669576111453616616, −3.75313942168689720041166296671, −2.24045872319457536057613420701, −1.18152064925248947837004719798,
1.26990601036553926798128041157, 2.52660248627938250576905248945, 3.36504896752369554051610435511, 4.57492324675834121825223765147, 5.20732753230937849669996872415, 6.26784849691681676767382426068, 6.63852731943742718201619396407, 8.024350534442175993078598384048, 8.473137596244116041885483283870, 8.958941341863152503021898675991