L(s) = 1 | − 1.83·2-s + 3-s + 1.35·4-s + (0.538 − 2.17i)5-s − 1.83·6-s + (−0.996 + 2.45i)7-s + 1.17·8-s + 9-s + (−0.986 + 3.97i)10-s + (−2.65 + 1.99i)11-s + 1.35·12-s + 1.25i·13-s + (1.82 − 4.49i)14-s + (0.538 − 2.17i)15-s − 4.87·16-s − 0.900i·17-s + ⋯ |
L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.679·4-s + (0.240 − 0.970i)5-s − 0.748·6-s + (−0.376 + 0.926i)7-s + 0.415·8-s + 0.333·9-s + (−0.312 + 1.25i)10-s + (−0.799 + 0.600i)11-s + 0.392·12-s + 0.347i·13-s + (0.488 − 1.20i)14-s + (0.139 − 0.560i)15-s − 1.21·16-s − 0.218i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05089949216\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05089949216\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.538 + 2.17i)T \) |
| 7 | \( 1 + (0.996 - 2.45i)T \) |
| 11 | \( 1 + (2.65 - 1.99i)T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 13 | \( 1 - 1.25iT - 13T^{2} \) |
| 17 | \( 1 + 0.900iT - 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 - 0.648iT - 23T^{2} \) |
| 29 | \( 1 + 8.58iT - 29T^{2} \) |
| 31 | \( 1 + 0.163iT - 31T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + 0.600iT - 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 - 6.57iT - 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 1.42iT - 73T^{2} \) |
| 79 | \( 1 + 6.04iT - 79T^{2} \) |
| 83 | \( 1 + 6.80iT - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−9.370694622482163082234436802537, −8.575112504371002590923400820404, −8.186408844409027888050581200138, −7.26176703100403442796849490574, −6.17323710887110705505707140074, −5.05696283517429534768928765442, −4.18531150363000481361297528529, −2.48941502046519520921758054807, −1.75861662348002127289310851254, −0.03035390247510967166876566622,
1.63271350663430588102149570970, 2.87760398778076534159895695590, 3.76098348427145048766828502609, 5.10481948828175304649791676717, 6.65023963299403717503207954211, 6.93699798068449422392580743820, 8.128123596805854889648899271313, 8.322119727960707725369500735534, 9.525571124720218126155439203764, 10.13534362454577913554970523394