| L(s) = 1 | + 2.86·5-s + (−2.43 − 1.03i)7-s + 3.32i·11-s − 0.821i·13-s + 3.52·17-s + 5.25i·19-s + 5.12i·23-s + 3.22·25-s + 1.64i·29-s − 2.55i·31-s + (−6.98 − 2.96i)35-s − 8.93·37-s + 3.49·41-s + 0.161·43-s − 5.34·47-s + ⋯ |
| L(s) = 1 | + 1.28·5-s + (−0.920 − 0.391i)7-s + 1.00i·11-s − 0.227i·13-s + 0.853·17-s + 1.20i·19-s + 1.06i·23-s + 0.645·25-s + 0.305i·29-s − 0.458i·31-s + (−1.18 − 0.501i)35-s − 1.46·37-s + 0.546·41-s + 0.0245·43-s − 0.779·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876322005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876322005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.43 + 1.03i)T \) |
| good | 5 | \( 1 - 2.86T + 5T^{2} \) |
| 11 | \( 1 - 3.32iT - 11T^{2} \) |
| 13 | \( 1 + 0.821iT - 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 - 5.12iT - 23T^{2} \) |
| 29 | \( 1 - 1.64iT - 29T^{2} \) |
| 31 | \( 1 + 2.55iT - 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 0.161T + 43T^{2} \) |
| 47 | \( 1 + 5.34T + 47T^{2} \) |
| 53 | \( 1 - 3.28iT - 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 8.43iT - 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.68iT - 71T^{2} \) |
| 73 | \( 1 - 7.86iT - 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−9.117673759182142040339486384254, −8.037030456397459749301036292684, −7.26305788620555275232202076712, −6.58612711953300870242128751404, −5.72739021183795225546179776555, −5.31293915851049103333497344069, −4.04196159390263276846896198324, −3.26252767111316389268736316248, −2.16967813702543475395217811564, −1.28458130888012680358315141416,
0.59307647633699720773697149638, 1.99649355977867010398874918688, 2.85385110892925448808616867966, 3.60630497749808257096703724150, 4.98273666155388259885197590450, 5.55626751255773744378451639879, 6.44081065342764636528694092986, 6.67247467996789365429934992211, 7.932788778893536693675801037316, 8.858375953836607988861986690153
