L(s) = 1 | + (−0.5 + 0.866i)3-s + (3 − 1.73i)5-s + (−0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s − 5.19i·13-s + 3.46i·15-s + (−6 − 3.46i)17-s + (−3.5 − 6.06i)19-s + (−2 − 1.73i)21-s + (3.5 − 6.06i)25-s + 0.999·27-s + (2.5 − 4.33i)31-s + (3 − 1.73i)33-s + (3 + 8.66i)35-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (1.34 − 0.774i)5-s + (−0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s − 1.44i·13-s + 0.894i·15-s + (−1.45 − 0.840i)17-s + (−0.802 − 1.39i)19-s + (−0.436 − 0.377i)21-s + (0.700 − 1.21i)25-s + 0.192·27-s + (0.449 − 0.777i)31-s + (0.522 − 0.301i)33-s + (0.507 + 1.46i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.171623863\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171623863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \) |
| 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 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.5 + 7.79i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.401125670394863516360453610713, −8.774647574478406259152580327013, −8.104652258271756420031806969559, −6.58943109740842429512723640410, −5.92557343419238126302057919266, −5.16019223796695592025845711692, −4.72163424407434968305944362402, −2.90449079513734976829214785036, −2.30824904945924204827088617707, −0.45446641738593379963099074030,
1.76067376641368651580485753200, 2.27417730767535000593169233888, 3.80968836216855497602501613923, 4.78941484943809288376211152268, 5.98713988355744226536585042432, 6.57688822912740895268988219936, 7.06514296176328701959695351664, 8.111294598068141072307414067493, 9.138583539104255913167845357693, 10.01358578187678467973285965681