L(s) = 1 | + (0.969 + 1.21i)3-s + (−1.39 + 1.10i)5-s + (−0.267 + 2.63i)7-s + (0.129 − 0.568i)9-s + (−2.43 + 0.556i)11-s + (−5.14 + 1.17i)13-s + (−2.69 − 0.615i)15-s + (0.0272 − 0.0565i)17-s − 1.00·19-s + (−3.45 + 2.22i)21-s + (−0.107 − 0.222i)23-s + (−0.409 + 1.79i)25-s + (5.01 − 2.41i)27-s + (−4.86 − 2.34i)29-s + 5.33·31-s + ⋯ |
L(s) = 1 | + (0.559 + 0.701i)3-s + (−0.621 + 0.495i)5-s + (−0.101 + 0.994i)7-s + (0.0432 − 0.189i)9-s + (−0.734 + 0.167i)11-s + (−1.42 + 0.325i)13-s + (−0.695 − 0.158i)15-s + (0.00661 − 0.0137i)17-s − 0.229·19-s + (−0.754 + 0.485i)21-s + (−0.0223 − 0.0463i)23-s + (−0.0818 + 0.358i)25-s + (0.965 − 0.465i)27-s + (−0.902 − 0.434i)29-s + 0.958·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.105459 + 0.894719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105459 + 0.894719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.267 - 2.63i)T \) |
good | 3 | \( 1 + (-0.969 - 1.21i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (1.39 - 1.10i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (2.43 - 0.556i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (5.14 - 1.17i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.0272 + 0.0565i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + (0.107 + 0.222i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (4.86 + 2.34i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 + (2.06 + 0.996i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (2.80 - 2.23i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.83 - 1.45i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.40 - 10.5i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (3.85 - 1.85i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-3.91 + 4.90i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (5.94 - 12.3i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 5.43iT - 67T^{2} \) |
| 71 | \( 1 + (1.70 + 3.54i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-5.36 - 1.22i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 - 14.8iT - 79T^{2} \) |
| 83 | \( 1 + (0.714 - 3.13i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-7.71 - 1.75i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 - 4.97iT - 97T^{2} \) |
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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
−10.54321005127287562079976500957, −9.672670946616023128718435680345, −9.195638633026442606492468672346, −8.142256812948628365489376649642, −7.41073370489227063898917457919, −6.35156237596904016300868156914, −5.16962033308962331511607439070, −4.28397794327762910595804155467, −3.14052154477356140070073442239, −2.38340674274045254218075020496,
0.39548552660413073963969667275, 2.03847025165802162020221946751, 3.21754379173991017882481954903, 4.45365593827869867655423975064, 5.23845013734370048958183690609, 6.75785820568334731238475508348, 7.54315439123589365715060704608, 7.941817055084748820687287065102, 8.817391047475982542209281219980, 10.05393534523976941803739613567