L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (1 − 1.73i)5-s − 7.99·8-s + (−1.99 − 3.46i)10-s + (−4 − 6.92i)11-s − 42·13-s + (−8 + 13.8i)16-s + (−1 − 1.73i)17-s + (62 − 107. i)19-s − 7.99·20-s − 15.9·22-s + (38 − 65.8i)23-s + (60.5 + 104. i)25-s + (−42 + 72.7i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0894 − 0.154i)5-s − 0.353·8-s + (−0.0632 − 0.109i)10-s + (−0.109 − 0.189i)11-s − 0.896·13-s + (−0.125 + 0.216i)16-s + (−0.0142 − 0.0247i)17-s + (0.748 − 1.29i)19-s − 0.0894·20-s − 0.155·22-s + (0.344 − 0.596i)23-s + (0.483 + 0.838i)25-s + (−0.316 + 0.548i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (4 + 6.92i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 42T + 2.19e3T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-62 + 107. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-38 + 65.8i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 254T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-36 - 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (199 - 344. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 462T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212T + 7.95e4T^{2} \) |
| 47 | \( 1 + (132 - 228. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (81 + 140. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (386 + 668. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (15 - 25.9i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-382 - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 236T + 3.57e5T^{2} \) |
| 73 | \( 1 + (209 + 361. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (276 - 478. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-15 + 25.9i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.348059075884290558662044359225, −8.532536226072913250494392903732, −7.36567973164797967155498064495, −6.58568620512480671500002888223, −5.23424739091545880754349025379, −4.86326167096767896252400760059, −3.48804790107523701954512735509, −2.62852374635398019471148225153, −1.36187223808993675589383215702, 0,
1.83686934965998388344058772584, 3.15316868092504720130332980679, 4.13769172186625743973252755457, 5.25050797376140655598605551675, 5.86329474625193205020151169480, 7.06702745227632497344772886662, 7.54268344911022424217676066981, 8.516685608785929237765059703184, 9.492125427186425551819928777797