L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + 0.999·20-s − 0.999·22-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + 0.999·20-s − 0.999·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7762427453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7762427453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.11649467267522829734197393932, −9.460920323510715531673400701629, −8.820130259655560051029766663259, −7.75414314137893028240538963046, −7.32380998864990173117437317453, −6.34137612611786791548647542959, −5.30194883798291579962600069860, −4.33627918375669845845352337523, −3.53830671774421151814635561205, −2.10145150397294409097414992252,
0.796102723187690449342926660761, 1.92269843499679796701520053419, 3.22029161399103696854180876123, 3.95038864210435977563621553114, 5.16674276485991631866735976195, 6.42905678795096232028728501384, 7.48543079372627703802208122715, 8.123674128320539566041894563446, 8.907037776560684756546142192240, 9.185071813372851386745157497706