L(s) = 1 | + (0.757 − 1.19i)2-s + (−0.817 + 3.05i)3-s + (−0.851 − 1.80i)4-s + (0.797 − 0.213i)5-s + (3.02 + 3.28i)6-s + (−2.80 − 0.353i)8-s + (−6.05 − 3.49i)9-s + (0.349 − 1.11i)10-s + (−0.732 + 2.73i)11-s + (6.22 − 1.12i)12-s + (−2.91 + 2.91i)13-s + 2.61i·15-s + (−2.54 + 3.08i)16-s + (−2.30 + 1.33i)17-s + (−8.75 + 4.57i)18-s + (0.436 − 0.117i)19-s + ⋯ |
L(s) = 1 | + (0.535 − 0.844i)2-s + (−0.472 + 1.76i)3-s + (−0.425 − 0.904i)4-s + (0.356 − 0.0956i)5-s + (1.23 + 1.34i)6-s + (−0.992 − 0.125i)8-s + (−2.01 − 1.16i)9-s + (0.110 − 0.352i)10-s + (−0.220 + 0.823i)11-s + (1.79 − 0.323i)12-s + (−0.807 + 0.807i)13-s + 0.673i·15-s + (−0.637 + 0.770i)16-s + (−0.558 + 0.322i)17-s + (−2.06 + 1.07i)18-s + (0.100 − 0.0268i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.272272 + 0.683293i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.272272 + 0.683293i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.757 + 1.19i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.817 - 3.05i)T + (-2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.797 + 0.213i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.732 - 2.73i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.91 - 2.91i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.30 - 1.33i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.436 + 0.117i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.63 + 2.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.04 + 2.04i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.26 - 2.18i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.625 - 2.33i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + (3.27 + 3.27i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.98 - 8.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.24 - 0.869i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.66 - 1.51i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.65 - 6.18i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-5.97 - 1.60i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 + (3.49 + 6.05i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.72 + 5.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.39 - 5.39i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.528 - 0.915i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−10.45207806180522738670963041796, −9.965435484160823972899228980897, −9.411603319747126775679701559606, −8.613856912432413464392527279059, −6.79555020049039509426880423873, −5.74597503826653696751455503044, −4.85311839513819026369464379582, −4.43624966975460904168567067701, −3.39010969352119713031620139862, −2.12527415568764995133882685355,
0.30384422903019000765024183292, 2.17501708447494291367496481997, 3.27678644258644470183891351216, 5.12769192676316787587273273707, 5.66035954236764711997959600701, 6.56032251157809215722211592536, 7.16544433118048717105494279166, 7.998285282005206860883255620420, 8.571292192155519916391081804787, 9.893837754602866941882744911062