L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−1.22 + 4.57i)7-s + (−0.707 − 0.707i)8-s + (−3 + 1.73i)11-s + (1.22 + 4.57i)13-s + (2.36 − 4.09i)14-s + (0.500 + 0.866i)16-s − 3.19i·19-s + (3.34 − 0.896i)22-s + (−2.12 + 0.568i)23-s − 4.73i·26-s + (−3.34 + 3.34i)28-s + (−5.36 − 9.29i)29-s + (−0.0980 + 0.169i)31-s + (−0.258 − 0.965i)32-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.462 + 1.72i)7-s + (−0.249 − 0.249i)8-s + (−0.904 + 0.522i)11-s + (0.339 + 1.26i)13-s + (0.632 − 1.09i)14-s + (0.125 + 0.216i)16-s − 0.733i·19-s + (0.713 − 0.191i)22-s + (−0.442 + 0.118i)23-s − 0.928i·26-s + (−0.632 + 0.632i)28-s + (−0.996 − 1.72i)29-s + (−0.0176 + 0.0305i)31-s + (−0.0457 − 0.170i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2939409824\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2939409824\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.22 - 4.57i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 4.57i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 3.19iT - 19T^{2} \) |
| 23 | \( 1 + (2.12 - 0.568i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (5.36 + 9.29i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.0980 - 0.169i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.79 + 5.79i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.448 + 0.120i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-5.79 - 1.55i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (5.79 + 5.79i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.76 + 4.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.34 - 1.43i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.26iT - 71T^{2} \) |
| 73 | \( 1 + (-3.67 + 3.67i)T - 73iT^{2} \) |
| 79 | \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.45 - 16.6i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 + (0.688 - 2.56i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−9.663775270489856846359848427486, −9.332026912636636638204257824929, −8.584252363737821281563148820570, −7.76710057731950592428046374949, −6.78583174971191218695236456347, −5.98095560145486577577873939680, −5.15374232047451666343380006853, −3.88088740379400647822316483180, −2.53177555171833949761184340466, −2.03722361217036383588523790545,
0.15144961281117062870837990593, 1.33707897316844174762692793249, 3.05536108466687217058505040237, 3.76957512766222212983511541904, 5.12098019161777128390365572752, 5.99192845347659700303909460155, 6.95885431886463964334121546336, 7.67390808631296919423695604742, 8.185205770189751672523797499303, 9.208472900816428322502391291632