L(s) = 1 | + (1 − 1.73i)5-s + 3·7-s + (−2.5 − 4.33i)13-s + (−2 + 3.46i)17-s + (4 − 1.73i)19-s + (−3 − 5.19i)23-s + (0.500 + 0.866i)25-s + (−2 − 3.46i)29-s + 7·31-s + (3 − 5.19i)35-s − 37-s + (5.5 − 9.52i)43-s + (−3 − 5.19i)47-s + 2·49-s + (−1 − 1.73i)53-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + 1.13·7-s + (−0.693 − 1.20i)13-s + (−0.485 + 0.840i)17-s + (0.917 − 0.397i)19-s + (−0.625 − 1.08i)23-s + (0.100 + 0.173i)25-s + (−0.371 − 0.643i)29-s + 1.25·31-s + (0.507 − 0.878i)35-s − 0.164·37-s + (0.838 − 1.45i)43-s + (−0.437 − 0.757i)47-s + 0.285·49-s + (−0.137 − 0.237i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.049192051\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.049192051\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.5 - 7.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.511535382787352037929656385904, −8.056498021957868892901208731156, −7.29647068910001796852401970758, −6.20876152325514070858790975166, −5.36491778444830718988827758542, −4.88968745443964316322907829406, −4.03989490455344729230709226881, −2.72277130391418595595419827095, −1.79780884082885014557631323343, −0.67347671895627927502237874484,
1.42155051377593773348860172898, 2.29204719660334904828069524856, 3.21976151845715936982518385080, 4.47351210385094920700078117889, 4.97222143434921049825181007593, 6.01513035474501692563058594377, 6.74792265808836219270128844740, 7.54427566591219358721419870725, 8.034341938820332371636715427319, 9.281968304190846881382656072117