L(s) = 1 | + (2.82 − 2.82i)2-s − 16.0i·4-s + (117. + 117. i)7-s + (−45.2 − 45.2i)8-s + 28.2i·11-s + (283. − 283. i)13-s + 662.·14-s − 256.·16-s + (−137. + 137. i)17-s + 2.80e3i·19-s + (79.9 + 79.9i)22-s + (−902. − 902. i)23-s − 1.60e3i·26-s + (1.87e3 − 1.87e3i)28-s − 827.·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.903 + 0.903i)7-s + (−0.250 − 0.250i)8-s + 0.0704i·11-s + (0.464 − 0.464i)13-s + 0.903·14-s − 0.250·16-s + (−0.115 + 0.115i)17-s + 1.78i·19-s + (0.0352 + 0.0352i)22-s + (−0.355 − 0.355i)23-s − 0.464i·26-s + (0.451 − 0.451i)28-s − 0.182·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.785731383\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.785731383\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-117. - 117. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 28.2iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-283. + 283. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (137. - 137. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.80e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (902. + 902. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 827.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.32e3 + 1.32e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.07e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (7.83e3 - 7.83e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.17e4 - 1.17e4i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.25e4 - 2.25e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.74e4 - 2.74e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.46e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.42e4 + 1.42e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 7.92e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.65e4 - 3.65e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 6.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.93e4 - 6.93e4i)T + 8.58e9iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52308389139798042302657030596, −9.643892377199913494212614141604, −8.484026644863968036403338050787, −7.87509033228796531842860589951, −6.29875380511372069098428514663, −5.57079064002882695917974487695, −4.58571919589077194411074549567, −3.45044364606795019870937483357, −2.21718892322195690087519913446, −1.25366592607995799248318913360,
0.57190479825520212918183048444, 2.02234868752179117218272074664, 3.53439811493672210751076336367, 4.50755013632540649148663228833, 5.28346878729467485170293515430, 6.62286638969943723569664894506, 7.26471289563970018510286696167, 8.254796423121522588864062987752, 9.082061456047929432950677598052, 10.33517403182305334299687132062