L(s) = 1 | + (−2.24 − 1.72i)2-s + (2.12 − 2.12i)3-s + (2.04 + 7.73i)4-s + (14.6 + 14.6i)5-s + (−8.41 + 1.09i)6-s − 26.8i·7-s + (8.77 − 20.8i)8-s − 8.99i·9-s + (−7.52 − 57.9i)10-s + (23.1 + 23.1i)11-s + (20.7 + 12.0i)12-s + (13.0 − 13.0i)13-s + (−46.4 + 60.2i)14-s + 61.9·15-s + (−55.6 + 31.5i)16-s − 5.45·17-s + ⋯ |
L(s) = 1 | + (−0.792 − 0.610i)2-s + (0.408 − 0.408i)3-s + (0.255 + 0.966i)4-s + (1.30 + 1.30i)5-s + (−0.572 + 0.0743i)6-s − 1.45i·7-s + (0.387 − 0.921i)8-s − 0.333i·9-s + (−0.237 − 1.83i)10-s + (0.635 + 0.635i)11-s + (0.498 + 0.290i)12-s + (0.278 − 0.278i)13-s + (−0.885 + 1.15i)14-s + 1.06·15-s + (−0.869 + 0.493i)16-s − 0.0778·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.20568 - 0.414186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20568 - 0.414186i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.24 + 1.72i)T \) |
| 3 | \( 1 + (-2.12 + 2.12i)T \) |
good | 5 | \( 1 + (-14.6 - 14.6i)T + 125iT^{2} \) |
| 7 | \( 1 + 26.8iT - 343T^{2} \) |
| 11 | \( 1 + (-23.1 - 23.1i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-13.0 + 13.0i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 5.45T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-4.68 + 4.68i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 34.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (143. - 143. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 97.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + (268. + 268. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 115. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-73.4 - 73.4i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 583.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-163. - 163. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (45.5 + 45.5i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-187. + 187. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (223. - 223. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 779. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 34.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 234.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-34.3 + 34.3i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.12e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−14.71154308193631228214265636631, −13.80639407176524079677525546884, −12.86812660695116072591558686668, −11.05420993181148112882186720721, −10.28247348411076936661848326793, −9.299291539963330267089311641982, −7.41673339320383708157434826325, −6.66836229406828240947047267616, −3.47151518336491235875506231898, −1.75318747837500017851449748052,
1.84743615602406385314155691345, 5.21523567042884923232709433863, 6.10812247890403782853126200158, 8.508266256313783172881955812721, 9.030489912811576477303901695052, 9.868463872006019767064443290584, 11.68043146734108963061509039687, 13.25188464711271354609377251716, 14.32676780475899997250292967494, 15.53639718936071689383327086205