| L(s) = 1 | + (−1 − 1.41i)3-s + 2.82·5-s + i·7-s + (−1.00 + 2.82i)9-s + 5.65i·11-s + 4i·13-s + (−2.82 − 4.00i)15-s + 2.82i·17-s + 2·19-s + (1.41 − i)21-s − 2.82·23-s + 3.00·25-s + (5.00 − 1.41i)27-s − 5.65·29-s + (8.00 − 5.65i)33-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s + 1.26·5-s + 0.377i·7-s + (−0.333 + 0.942i)9-s + 1.70i·11-s + 1.10i·13-s + (−0.730 − 1.03i)15-s + 0.685i·17-s + 0.458·19-s + (0.308 − 0.218i)21-s − 0.589·23-s + 0.600·25-s + (0.962 − 0.272i)27-s − 1.05·29-s + (1.39 − 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33913 + 0.425627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33913 + 0.425627i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.56472840459288903269395088855, −9.678586792387708958396001594294, −9.089345299890934325656673945828, −7.72680082531768232915413534229, −6.96540789622041837596297961291, −6.10392986975012395256590188592, −5.40813517959493494027930335587, −4.29793808407072369004329844798, −2.20849493224236845721345450712, −1.76649967739551748134831501980,
0.825030282542565167897527962824, 2.80831787735347733612055588396, 3.81009593026033240300305234475, 5.30958829214338860243821758092, 5.67604898182184326782463530262, 6.52732043184584880743036062622, 7.907300913890604474270513112213, 8.974225825684200794269720092086, 9.684504368472198366479724420363, 10.42817144679775522868303161433
