L(s) = 1 | + (0.448 − 0.258i)2-s + (−0.448 − 1.67i)3-s + (−0.866 + 1.5i)4-s + (2.09 − 0.792i)5-s + (−0.633 − 0.633i)6-s + (−2.89 + 1.67i)7-s + 1.93i·8-s + (−2.59 + 1.50i)9-s + (0.732 − 0.896i)10-s + (−0.633 − 1.09i)11-s + (2.89 + 0.776i)12-s + (2.12 + 1.22i)13-s + (−0.866 + 1.5i)14-s + (−2.26 − 3.14i)15-s + (−1.23 − 2.13i)16-s − 5.27i·17-s + ⋯ |
L(s) = 1 | + (0.316 − 0.183i)2-s + (−0.258 − 0.965i)3-s + (−0.433 + 0.750i)4-s + (0.935 − 0.354i)5-s + (−0.258 − 0.258i)6-s + (−1.09 + 0.632i)7-s + 0.683i·8-s + (−0.866 + 0.5i)9-s + (0.231 − 0.283i)10-s + (−0.191 − 0.331i)11-s + (0.836 + 0.224i)12-s + (0.588 + 0.339i)13-s + (−0.231 + 0.400i)14-s + (−0.584 − 0.811i)15-s + (−0.308 − 0.533i)16-s − 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.799798 - 0.208566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799798 - 0.208566i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.448 + 1.67i)T \) |
| 5 | \( 1 + (-2.09 + 0.792i)T \) |
good | 2 | \( 1 + (-0.448 + 0.258i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (2.89 - 1.67i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.27iT - 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 + (0.448 + 0.258i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.232 + 0.401i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.366 - 0.633i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.568 - 0.328i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.56 + 1.48i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.03iT - 53T^{2} \) |
| 59 | \( 1 + (-4.73 + 8.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.33 - 5.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.57 + 3.79i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 + (-3.73 - 6.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.90 - 3.98i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-13.1 + 7.58i)T + (48.5 - 84.0i)T^{2} \) |
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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
−16.13652009346099422441644688589, −14.00015829763810068807773623996, −13.33593292676355094284024557067, −12.56405101840455539096303402126, −11.52524957482138386805832242667, −9.511776945928018045698065455111, −8.420197340438452041944132623096, −6.67073109823986074166042756097, −5.36138558065511476615761428426, −2.80491962154153509206262157101,
3.79044380172160831923546015431, 5.53845487395368494958012742528, 6.49555265506791225858237422666, 9.113088169207754924920855174079, 10.14764321295442064126155322479, 10.63464342017533185789574606221, 12.85874656992119752403751633841, 13.80884469481467588592705882169, 14.86534726700461892470528417738, 15.81056797997840752100286058482