L(s) = 1 | + (1 + 1.73i)4-s + (−0.5 − 2.59i)7-s − 7·13-s + (−1.99 + 3.46i)16-s + (3.5 − 6.06i)19-s + (2.5 + 4.33i)25-s + (4 − 3.46i)28-s + (3.5 + 6.06i)31-s + (0.5 − 0.866i)37-s + 5·43-s + (−6.5 + 2.59i)49-s + (−7 − 12.1i)52-s + (−7 + 12.1i)61-s − 7.99·64-s + (−5.5 − 9.52i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (−0.188 − 0.981i)7-s − 1.94·13-s + (−0.499 + 0.866i)16-s + (0.802 − 1.39i)19-s + (0.5 + 0.866i)25-s + (0.755 − 0.654i)28-s + (0.628 + 1.08i)31-s + (0.0821 − 0.142i)37-s + 0.762·43-s + (−0.928 + 0.371i)49-s + (−0.970 − 1.68i)52-s + (−0.896 + 1.55i)61-s − 0.999·64-s + (−0.671 − 1.16i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.916115 + 0.116743i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.916115 + 0.116743i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−15.10391512386323464488553043475, −13.81857004699226797923647639741, −12.75045934194220757550721874333, −11.76131590485914725834823505798, −10.57513541047567563154004230669, −9.250577143948027650400233365546, −7.56481020328747985759872945590, −6.95697683731745953258666267602, −4.71954527606832227016107159929, −2.95585781887631646524441704782,
2.45311402063529036694048344147, 5.06428004548084692695021768767, 6.22998923426805380952819476845, 7.70102303608271730101848054391, 9.438431329900591574938370924126, 10.20281678875964561378998763774, 11.73034383769830250715065081985, 12.43916690255645319183688183218, 14.19759421423389586027241204546, 14.89204274700472610334583207136