L(s) = 1 | + (−1.57 + 1.97i)2-s + (−0.0816 + 0.357i)3-s + (−0.971 − 4.25i)4-s + (−0.577 − 0.724i)6-s + (−4.99 − 1.14i)7-s + (5.37 + 2.59i)8-s + (2.58 + 1.24i)9-s + (−1.11 − 2.31i)11-s + 1.60·12-s + (0.481 + 0.999i)13-s + (10.1 − 8.06i)14-s + (−5.70 + 2.74i)16-s + 1.36·17-s + (−6.51 + 3.13i)18-s + (−1.06 + 0.242i)19-s + ⋯ |
L(s) = 1 | + (−1.11 + 1.39i)2-s + (−0.0471 + 0.206i)3-s + (−0.485 − 2.12i)4-s + (−0.235 − 0.295i)6-s + (−1.88 − 0.431i)7-s + (1.90 + 0.915i)8-s + (0.860 + 0.414i)9-s + (−0.336 − 0.698i)11-s + 0.462·12-s + (0.133 + 0.277i)13-s + (2.70 − 2.15i)14-s + (−1.42 + 0.686i)16-s + 0.332·17-s + (−1.53 + 0.739i)18-s + (−0.244 + 0.0556i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.374153 + 0.471345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.374153 + 0.471345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 + (5.04 - 1.87i)T \) |
good | 2 | \( 1 + (1.57 - 1.97i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (0.0816 - 0.357i)T + (-2.70 - 1.30i)T^{2} \) |
| 7 | \( 1 + (4.99 + 1.14i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (1.11 + 2.31i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.481 - 0.999i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 19 | \( 1 + (1.06 - 0.242i)T + (17.1 - 8.24i)T^{2} \) |
| 23 | \( 1 + (-6.40 + 5.10i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-5.93 - 4.73i)T + (6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 1.32i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 - 7.19iT - 41T^{2} \) |
| 43 | \( 1 + (-3.50 - 4.39i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-7.68 + 3.70i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-2.87 - 2.29i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + (0.592 + 0.135i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (0.977 - 2.03i)T + (-41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (-8.05 + 3.87i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.86 - 2.33i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + (-4.20 + 8.74i)T + (-49.2 - 61.7i)T^{2} \) |
| 83 | \( 1 + (10.1 - 2.31i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-4.10 - 3.27i)T + (19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (-1.80 - 7.91i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.35316358815101771465178441187, −9.552812989879075894783060611785, −9.003071159588579976797834937285, −7.983587244506006640821401469332, −7.04804408578449330435901232350, −6.56582540139543688044308846993, −5.73181988860055342281840541699, −4.51404663956215343565309803229, −3.09583129941502234722479378899, −0.846559459599159057999211386670,
0.68673381258030327082055936307, 2.23040067872798139086571524595, 3.19936085757397385150237717022, 4.04294881216052520894559923121, 5.81082813645612326474111122764, 6.99524008648836999959750505318, 7.65258663747511978875586835065, 8.996975714674478303687558839053, 9.490655442523023690124103856147, 9.998738589667151264191493685534