L(s) = 1 | + (−1.05 − 0.946i)2-s + (0.207 + 1.98i)4-s − 1.60i·5-s + (1.66 − 2.28i)8-s + (−1.52 + 1.68i)10-s + 2.67i·11-s + 3.37i·13-s + (−3.91 + 0.823i)16-s − 1.60i·17-s − 4.30·19-s + (3.19 − 0.333i)20-s + (2.53 − 2.81i)22-s − 6.46i·23-s + 2.41·25-s + (3.19 − 3.54i)26-s + ⋯ |
L(s) = 1 | + (−0.742 − 0.669i)2-s + (0.103 + 0.994i)4-s − 0.719i·5-s + (0.588 − 0.808i)8-s + (−0.481 + 0.534i)10-s + 0.807i·11-s + 0.937i·13-s + (−0.978 + 0.205i)16-s − 0.390i·17-s − 0.987·19-s + (0.715 − 0.0744i)20-s + (0.540 − 0.599i)22-s − 1.34i·23-s + 0.482·25-s + (0.627 − 0.696i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4004465498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4004465498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.05 + 0.946i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.60iT - 5T^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 1.60iT - 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 + 6.46iT - 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 + 11.6iT - 41T^{2} \) |
| 43 | \( 1 + 7.95iT - 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 + 9.04T + 59T^{2} \) |
| 61 | \( 1 + 3.37iT - 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 8.03iT - 71T^{2} \) |
| 73 | \( 1 + 6.17iT - 73T^{2} \) |
| 79 | \( 1 - 3.29iT - 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 0.275iT - 89T^{2} \) |
| 97 | \( 1 + 12.9iT - 97T^{2} \) |
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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
−8.921383598568056637871396381198, −8.514214167481470088458637435782, −7.25842843177226665740718228987, −6.94743381340757434684565858976, −5.56203853567446445329648120635, −4.44154840689863338478919452999, −3.92770727777729797439452397112, −2.44026858045508042386082602430, −1.69592881644758942337173436844, −0.19218112262037312034461936932,
1.41807147242628669580759384893, 2.74746829229275986244338460091, 3.78919641479799300272165655378, 5.14946993899483747619889937149, 5.90751954937582563562301368357, 6.51029265074244070887682033564, 7.53435027845827727217944983939, 7.937623720113277097167391184740, 8.925612128149389243726696005139, 9.506758426588787569950933777218