L(s) = 1 | − 5.07i·2-s − 17.7·4-s + (−4.21 + 7.29i)5-s + (−4.29 + 18.0i)7-s + 49.6i·8-s + (37.0 + 21.3i)10-s + (39.7 − 22.9i)11-s + (18.3 − 10.6i)13-s + (91.4 + 21.8i)14-s + 109.·16-s + (8.26 − 14.3i)17-s + (−49.7 + 28.7i)19-s + (74.8 − 129. i)20-s + (−116. − 201. i)22-s + (167. + 96.8i)23-s + ⋯ |
L(s) = 1 | − 1.79i·2-s − 2.22·4-s + (−0.376 + 0.652i)5-s + (−0.232 + 0.972i)7-s + 2.19i·8-s + (1.17 + 0.676i)10-s + (1.08 − 0.628i)11-s + (0.392 − 0.226i)13-s + (1.74 + 0.416i)14-s + 1.71·16-s + (0.117 − 0.204i)17-s + (−0.600 + 0.346i)19-s + (0.837 − 1.45i)20-s + (−1.12 − 1.95i)22-s + (1.52 + 0.877i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.17126 - 0.480092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17126 - 0.480092i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (4.29 - 18.0i)T \) |
good | 2 | \( 1 + 5.07iT - 8T^{2} \) |
| 5 | \( 1 + (4.21 - 7.29i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-39.7 + 22.9i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-18.3 + 10.6i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-8.26 + 14.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (49.7 - 28.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-167. - 96.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-47.1 - 27.2i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 294. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (166. + 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-275. - 159. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 298.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 226. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 99.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 176. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-142. - 82.2i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (457. - 792. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (67.8 + 117. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (377. + 217. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.76373535533439227702286758930, −11.21105800804125873151920168674, −10.30160039162990749317800669199, −9.145663376776289312836791848161, −8.559727024358292803302811425552, −6.70412470732800003832907184938, −5.19452453450264095681996590171, −3.61603829924361167674474358163, −2.93138458625007207555352374818, −1.30450405917732960993218469331,
0.67432804403257040026858539788, 4.09371015643040528859721392785, 4.68548795171869343779307752435, 6.27680190588581735976023193630, 6.95162669772234694329904526551, 7.985914440412572480115889972276, 8.894252291596093220815058767446, 9.752954166264982513484109156865, 11.25615434187081442443155411867, 12.75927531912731930049991514242