L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.328 + 1.22i)7-s + (0.707 + 0.707i)8-s + (−3 + 1.73i)11-s + (0.328 + 1.22i)13-s + (−0.633 + 1.09i)14-s + (0.500 + 0.866i)16-s + 7.19i·19-s + (−3.34 + 0.896i)22-s + (−7.91 + 2.12i)23-s + 1.26i·26-s + (−0.896 + 0.896i)28-s + (3.63 + 6.29i)29-s + (5.09 − 8.83i)31-s + (0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.124 + 0.462i)7-s + (0.249 + 0.249i)8-s + (−0.904 + 0.522i)11-s + (0.0910 + 0.339i)13-s + (−0.169 + 0.293i)14-s + (0.125 + 0.216i)16-s + 1.65i·19-s + (−0.713 + 0.191i)22-s + (−1.65 + 0.442i)23-s + 0.248i·26-s + (−0.169 + 0.169i)28-s + (0.674 + 1.16i)29-s + (0.915 − 1.58i)31-s + (0.0457 + 0.170i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.957301754\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.957301754\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.328 - 1.22i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.328 - 1.22i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 7.19iT - 19T^{2} \) |
| 23 | \( 1 + (7.91 - 2.12i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.63 - 6.29i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.09 + 8.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.24 - 1.67i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.55 + 1.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.23 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.0 - 3.22i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 10.7iT - 71T^{2} \) |
| 73 | \( 1 + (3.67 - 3.67i)T - 73iT^{2} \) |
| 79 | \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.76 + 6.57i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 + (-3.79 + 14.1i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.03769061700729037827820474792, −8.999795572715154623698515797333, −7.924353909829294984201078545005, −7.57150005971787929440952364090, −6.20415113869645550855073121373, −5.85366134951655518247665950931, −4.75068985588821584343728116911, −3.94065396220301395671835178167, −2.80971730472912072711294065948, −1.80400196965886297800308886043,
0.60993680649904455083546287483, 2.34113740123553815283726734825, 3.16107108757377568312868284609, 4.28731146710516578712095962240, 5.02333357849803161382080547452, 6.01800675616353995101855000438, 6.72869053890049284883246563448, 7.74069728023359053631134758569, 8.410966246024981727403659907251, 9.525639604910268033674257705632