L(s) = 1 | + (−3.10 − 1.79i)2-s + (2.44 + 4.22i)4-s + (−0.428 + 0.742i)5-s + 11.1i·8-s + (2.66 − 1.53i)10-s + (0.321 − 0.185i)11-s − 62.0i·13-s + (39.6 − 68.6i)16-s + (51.9 + 90.0i)17-s + (−61.9 − 35.7i)19-s − 4.18·20-s − 1.33·22-s + (−118. − 68.2i)23-s + (62.1 + 107. i)25-s + (−111. + 192. i)26-s + ⋯ |
L(s) = 1 | + (−1.09 − 0.634i)2-s + (0.305 + 0.528i)4-s + (−0.0383 + 0.0664i)5-s + 0.494i·8-s + (0.0842 − 0.0486i)10-s + (0.00880 − 0.00508i)11-s − 1.32i·13-s + (0.618 − 1.07i)16-s + (0.741 + 1.28i)17-s + (−0.747 − 0.431i)19-s − 0.0468·20-s − 0.0129·22-s + (−1.07 − 0.618i)23-s + (0.497 + 0.860i)25-s + (−0.839 + 1.45i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4623948282\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4623948282\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (3.10 + 1.79i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (0.428 - 0.742i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.321 + 0.185i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-51.9 - 90.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (61.9 + 35.7i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (118. + 68.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 17.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (93.2 - 53.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-190. + 330. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (261. - 452. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (268. - 154. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-174. - 302. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-580. - 335. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-119. - 207. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 178. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-117. + 67.7i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (428. - 742. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-282. + 489. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.73e3iT - 9.12e5T^{2} \) |
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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
−10.58032320903922495571827784420, −10.17006214152529484556110799735, −9.107318258385971302821925355931, −8.266571616150628613396519561565, −7.64352928931486828307852644271, −6.18146463218450013112278974231, −5.18520960062107756485426099441, −3.63137262371067659583727664195, −2.37758769744318071138425511418, −1.09203739587861223592899981840,
0.25340278057354898394157614318, 1.80418043499802082622461527346, 3.58371983455573300388031787183, 4.80864248109927875998223335558, 6.25958702714163973283051340145, 6.94360590862543850277851140887, 7.953270274827444561930294638058, 8.611513889581856635842571429396, 9.673206428045912270865206392858, 10.01716584937061613970825087296