L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (−3.67 + 2.12i)5-s + 2.82i·8-s + (3 − 5.19i)10-s + (14.6 + 8.48i)11-s − 8·13-s + (−2.00 − 3.46i)16-s + (11.0 + 6.36i)17-s + (−8 − 13.8i)19-s + 8.48i·20-s − 24·22-s + (14.6 − 8.48i)23-s + (−3.5 + 6.06i)25-s + (9.79 − 5.65i)26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.734 + 0.424i)5-s + 0.353i·8-s + (0.300 − 0.519i)10-s + (1.33 + 0.771i)11-s − 0.615·13-s + (−0.125 − 0.216i)16-s + (0.648 + 0.374i)17-s + (−0.421 − 0.729i)19-s + 0.424i·20-s − 1.09·22-s + (0.638 − 0.368i)23-s + (−0.140 + 0.242i)25-s + (0.376 − 0.217i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8643851527\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8643851527\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (3.67 - 2.12i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-14.6 - 8.48i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 + (-11.0 - 6.36i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (8 + 13.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-14.6 + 8.48i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 + (-22 + 38.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-17 - 29.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + (73.4 - 42.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-33.0 - 19.0i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (29.3 + 16.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-25 - 43.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (8 - 13.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38 - 65.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-11.0 + 6.36i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 176T + 9.40e3T^{2} \) |
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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
−9.928113462314503542509869471272, −9.543704094549151842834607024469, −8.454187802152264854383250394307, −7.70766503201771374998511878492, −6.88685179351601789470443518664, −6.30094403737312104757359129181, −4.88923233735178366573834220899, −3.99551639365702219500021050689, −2.70082817516649637689024406973, −1.23593945395795087787291779759,
0.39739311974999042794728534271, 1.59101870576369595937035838314, 3.18961281053659149247840428680, 3.96506829180250120192089407213, 5.11393037739498814138157043515, 6.36437199675942058731137304594, 7.23165564165952919872077076216, 8.171580806826909560975255319066, 8.761408250986675665963867266210, 9.575975950268339628134603841893