L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (1.5 + 0.866i)11-s + (−1.79 + 6.69i)13-s + (0.500 − 0.866i)16-s + (2.12 + 2.12i)17-s + 4i·19-s + (1.67 + 0.448i)22-s + (5.79 + 1.55i)23-s + 6.92i·26-s + (1.73 − 3i)29-s + (−2 − 3.46i)31-s + (0.258 − 0.965i)32-s + (2.59 + 1.5i)34-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.249 − 0.249i)8-s + (0.452 + 0.261i)11-s + (−0.497 + 1.85i)13-s + (0.125 − 0.216i)16-s + (0.514 + 0.514i)17-s + 0.917i·19-s + (0.356 + 0.0955i)22-s + (1.20 + 0.323i)23-s + 1.35i·26-s + (0.321 − 0.557i)29-s + (−0.359 − 0.622i)31-s + (0.0457 − 0.170i)32-s + (0.445 + 0.257i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.581365768\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.581365768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.79 - 6.69i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-5.79 - 1.55i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 3i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.36 - 2.24i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-11.5 + 3.10i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.48 - 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.34 + 0.896i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \) |
| 79 | \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.32 - 8.69i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 + (-4.03 - 15.0i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−9.547635406599191271947038492050, −9.133766080739873216869112470816, −7.86425901056988832122361274492, −7.07768202545864391097347777478, −6.31201267855521923735988308256, −5.44796441835116009432376673666, −4.35734087313443383930056255506, −3.83283042588742194556316194249, −2.47951830024708823576546083451, −1.46199607149745418142887689695,
0.937856155063854380064636364261, 2.76815434471160039885319125454, 3.26505955360919054124322772418, 4.66206539009355371446905469383, 5.25149498324785092658375942341, 6.14928024727211321810981990423, 7.09232916182077252045462630145, 7.75157445352009272002768304013, 8.672082223009893816065037526869, 9.541355737177464523077917398618