L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.404 + 0.339i)3-s + (−0.939 + 0.342i)4-s + (2.97 + 1.08i)5-s + (0.404 + 0.339i)6-s + (0.0238 − 0.0412i)7-s + (0.5 + 0.866i)8-s + (−0.472 + 2.67i)9-s + (0.550 − 3.12i)10-s + (0.5 + 0.866i)11-s + (0.264 − 0.457i)12-s + (−1.99 − 1.67i)13-s + (−0.0447 − 0.0162i)14-s + (−1.57 + 0.572i)15-s + (0.766 − 0.642i)16-s + (1.16 + 6.61i)17-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.233 + 0.196i)3-s + (−0.469 + 0.171i)4-s + (1.33 + 0.484i)5-s + (0.165 + 0.138i)6-s + (0.00900 − 0.0155i)7-s + (0.176 + 0.306i)8-s + (−0.157 + 0.893i)9-s + (0.174 − 0.987i)10-s + (0.150 + 0.261i)11-s + (0.0762 − 0.132i)12-s + (−0.553 − 0.464i)13-s + (−0.0119 − 0.00435i)14-s + (−0.406 + 0.147i)15-s + (0.191 − 0.160i)16-s + (0.283 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36628 + 0.169347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36628 + 0.169347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (1.25 - 4.17i)T \) |
good | 3 | \( 1 + (0.404 - 0.339i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.97 - 1.08i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.0238 + 0.0412i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (1.99 + 1.67i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 6.61i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.12 + 2.23i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.334 + 1.89i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.96 + 6.87i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.66T + 37T^{2} \) |
| 41 | \( 1 + (1.96 - 1.65i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (10.0 + 3.65i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.944 + 5.35i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.98 + 1.08i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.09 - 6.19i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.302 + 0.110i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.0865 + 0.490i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (12.4 + 4.52i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.76 - 3.15i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.86 + 2.40i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.22 + 12.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (13.4 + 11.2i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.59 - 14.7i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.97096030835111894993518424066, −10.18613013213638332578964260855, −9.973879199320264129707990122951, −8.664723351158138471156515471222, −7.69845113376213390003014250474, −6.26522070422305283969140301259, −5.50957831868983778725289544330, −4.32181288034016724799102106745, −2.76386130601538463073420325632, −1.77569216023920322787414806326,
1.05746268675895375007057451854, 2.88466540148894014634597219253, 4.78910724031869096357267541483, 5.45391876030049160895844249738, 6.56769841904920878512430115441, 7.09668042841005535738885083565, 8.654352163170768923897034975660, 9.346194606479386258178051788514, 9.813496807338083981887114276902, 11.21632825303891619247024249453