| L(s) = 1 | + (1.23 − 3.80i)2-s + (−12.9 − 9.40i)4-s + (−7.80 − 24.0i)5-s + (84.3 + 61.2i)7-s + (−51.7 + 37.6i)8-s − 101.·10-s + (−301. + 265. i)11-s + (144. − 445. i)13-s + (337. − 245. i)14-s + (79.1 + 243. i)16-s + (524. + 1.61e3i)17-s + (−1.73e3 + 1.26e3i)19-s + (−124. + 384. i)20-s + (637. + 1.47e3i)22-s − 2.08e3·23-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.139 − 0.429i)5-s + (0.650 + 0.472i)7-s + (−0.286 + 0.207i)8-s − 0.319·10-s + (−0.750 + 0.661i)11-s + (0.237 − 0.730i)13-s + (0.460 − 0.334i)14-s + (0.0772 + 0.237i)16-s + (0.439 + 1.35i)17-s + (−1.10 + 0.802i)19-s + (−0.0698 + 0.214i)20-s + (0.280 + 0.649i)22-s − 0.820·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.650885909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650885909\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (301. - 265. i)T \) |
| good | 5 | \( 1 + (7.80 + 24.0i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-84.3 - 61.2i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-144. + 445. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-524. - 1.61e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.73e3 - 1.26e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 2.08e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-4.30e3 - 3.12e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-193. + 596. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.32e4 - 9.60e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.56e4 - 1.13e4i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.40e3 + 1.74e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.69e3 + 8.30e3i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-6.64e3 - 4.82e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.58e4 - 4.89e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 6.44e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-7.85e3 - 2.41e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.04e4 - 2.93e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-4.82e3 + 1.48e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.78e4 + 8.56e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 523.T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.43e3 - 1.36e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−11.83795782160779617056030298184, −10.57760340715041410529887015117, −10.06439681334539372389468042618, −8.454957726087668516560735284234, −8.086957703318241455398984128050, −6.19250873514070922670460746462, −5.09972146160296856646717602842, −4.07622714160360131310388818614, −2.51968743561091045101290005189, −1.27668802185968960658112963288,
0.50889716352908287336754093119, 2.58710045955298018686153070206, 4.09389106758032434913434045273, 5.14763040331301939304677643066, 6.42474998759361881648331096068, 7.40230639242864872596017136135, 8.270611004520677880745389903643, 9.379707955378419808378173257944, 10.71900643742147061590537289274, 11.40342215344647146889876783203
