L(s) = 1 | + (2.27 − 1.68i)2-s + 4.93i·3-s + (2.33 − 7.65i)4-s − 13.0i·5-s + (8.29 + 11.2i)6-s + 6.24·7-s + (−7.56 − 21.3i)8-s + 2.68·9-s + (−21.8 − 29.5i)10-s − 46.1i·11-s + (37.7 + 11.5i)12-s + 58.0i·13-s + (14.1 − 10.5i)14-s + 64.1·15-s + (−53.0 − 35.7i)16-s + 17·17-s + ⋯ |
L(s) = 1 | + (0.803 − 0.594i)2-s + 0.949i·3-s + (0.292 − 0.956i)4-s − 1.16i·5-s + (0.564 + 0.762i)6-s + 0.336·7-s + (−0.334 − 0.942i)8-s + 0.0992·9-s + (−0.691 − 0.934i)10-s − 1.26i·11-s + (0.907 + 0.277i)12-s + 1.23i·13-s + (0.270 − 0.200i)14-s + 1.10·15-s + (−0.829 − 0.558i)16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.14780 - 1.51706i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14780 - 1.51706i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.27 + 1.68i)T \) |
| 17 | \( 1 - 17T \) |
good | 3 | \( 1 - 4.93iT - 27T^{2} \) |
| 5 | \( 1 + 13.0iT - 125T^{2} \) |
| 7 | \( 1 - 6.24T + 343T^{2} \) |
| 11 | \( 1 + 46.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 58.0iT - 2.19e3T^{2} \) |
| 19 | \( 1 + 154. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 230.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 110.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 477. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 205. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 242. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 30.7iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 696.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 841.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 752.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 302.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 104.T + 9.12e5T^{2} \) |
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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
−12.68186529464575034692653362148, −11.29987361899811384017325783374, −10.91278677930465635227730379830, −9.264535335496346209440639640472, −8.972473574888186744321597341474, −6.82196786586063685267032819051, −5.11630877086192130664887296643, −4.69903575969259158928835219845, −3.30899507964282465366996181909, −1.17688960427253693271996888084,
2.12391811338660214416554118459, 3.60242196421452703572159749580, 5.31176411079516859358615754436, 6.57737664379953200121568077695, 7.37564053152880112263914903213, 8.021729167897592068123816016251, 10.01360508514476306684469174769, 11.14863142282887277276595349703, 12.40706381008447061626460511989, 12.84398518446107975652553595646