| L(s) = 1 | − i·3-s + 0.646·5-s − 3.09i·7-s − 9-s − 1.41·11-s + (−2.44 + 2.64i)13-s − 0.646i·15-s − 5.58·17-s + 5.36·19-s − 3.09·21-s − 3.46·23-s − 4.58·25-s + i·27-s − 4.37i·29-s + 9.28i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.288·5-s − 1.17i·7-s − 0.333·9-s − 0.426·11-s + (−0.679 + 0.733i)13-s − 0.166i·15-s − 1.35·17-s + 1.23·19-s − 0.675·21-s − 0.722·23-s − 0.916·25-s + 0.192i·27-s − 0.812i·29-s + 1.66i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1252438032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1252438032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 13 | \( 1 + (2.44 - 2.64i)T \) |
| good | 5 | \( 1 - 0.646T + 5T^{2} \) |
| 7 | \( 1 + 3.09iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 19 | \( 1 - 5.36T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 4.37iT - 29T^{2} \) |
| 31 | \( 1 - 9.28iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 12.4iT - 41T^{2} \) |
| 43 | \( 1 - 5.58iT - 43T^{2} \) |
| 47 | \( 1 + 3.74iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 4.37iT - 61T^{2} \) |
| 67 | \( 1 + 4.77T + 67T^{2} \) |
| 71 | \( 1 + 2.44iT - 71T^{2} \) |
| 73 | \( 1 - 5.65iT - 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 - 5.06iT - 97T^{2} \) |
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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
−9.267485193249060989155163641209, −8.214216556229511776660767408456, −7.61586864768469518003601799270, −6.85945175422305119223399612715, −6.36532759924550909875595520703, −5.19320051658827487226788122529, −4.47802989642768604644722851625, −3.48174409638295681169038939380, −2.35104921133565807008292670298, −1.39593221325462316395424577816,
0.03904175705137962963122979607, 2.07038181056766066578783438650, 2.71286719908219367101281421278, 3.79676903864017581279761803916, 4.83631166405137332597443898631, 5.58322432766758298829908955665, 5.99211758675098134577901124164, 7.26491862334785797190837376066, 7.917105675089204723476855523998, 9.000675098725905267846832475413
