L(s) = 1 | + (1.40 − 0.167i)2-s + (1.94 − 0.470i)4-s + (0.707 − 1.70i)5-s + (−2.74 − 2.74i)7-s + (2.65 − 0.985i)8-s + (0.707 − 2.51i)10-s + (−0.135 − 0.0560i)11-s + (1.18 + 2.85i)13-s + (−4.32 − 3.40i)14-s + (3.55 − 1.82i)16-s + 6.44i·17-s + (0.805 + 1.94i)19-s + (0.571 − 3.65i)20-s + (−0.199 − 0.0560i)22-s + (−0.749 + 0.749i)23-s + ⋯ |
L(s) = 1 | + (0.992 − 0.118i)2-s + (0.971 − 0.235i)4-s + (0.316 − 0.763i)5-s + (−1.03 − 1.03i)7-s + (0.937 − 0.348i)8-s + (0.223 − 0.795i)10-s + (−0.0408 − 0.0169i)11-s + (0.327 + 0.790i)13-s + (−1.15 − 0.908i)14-s + (0.889 − 0.457i)16-s + 1.56i·17-s + (0.184 + 0.445i)19-s + (0.127 − 0.816i)20-s + (−0.0425 − 0.0119i)22-s + (−0.156 + 0.156i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07540 - 0.854344i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07540 - 0.854344i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.167i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.707 + 1.70i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (2.74 + 2.74i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.135 + 0.0560i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.18 - 2.85i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 6.44iT - 17T^{2} \) |
| 19 | \( 1 + (-0.805 - 1.94i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.749 - 0.749i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.32 - 1.79i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + (-1.73 + 4.18i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.49 + 2.49i)T - 41iT^{2} \) |
| 43 | \( 1 + (6.10 + 2.52i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 2.66iT - 47T^{2} \) |
| 53 | \( 1 + (1.64 + 0.682i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.43 - 3.47i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (3.46 - 1.43i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (14.0 - 5.83i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-3.40 - 3.40i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.442 - 0.442i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.07iT - 79T^{2} \) |
| 83 | \( 1 + (2.99 + 7.23i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.21 - 4.21i)T + 89iT^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−11.94659080905821666244046297708, −10.77176833474274350398555626721, −10.06903883068797716889164819913, −8.948319124319717679709438669539, −7.53645542044266062080830067920, −6.52231718111350978835705899887, −5.66528020989761744944015480842, −4.28721634869839379405815941781, −3.51217662958108961341824946449, −1.58740242097832837673522410638,
2.60884822454516106776966240064, 3.20374834788595716431735964001, 4.94527341390506713238655212494, 6.00141764304122725010045128828, 6.66308628609074644450148298920, 7.78193154552871334085420666282, 9.227866553197712331543422700423, 10.17137654863614522272949270604, 11.23056759697113292444149193189, 12.03526596845764263982398333377