L(s) = 1 | + (0.781 + 0.623i)2-s + (−0.864 + 1.79i)3-s + (0.222 + 0.974i)4-s + (2.20 + 0.383i)5-s + (−1.79 + 0.864i)6-s + (−2.09 + 1.61i)7-s + (−0.433 + 0.900i)8-s + (−0.605 − 0.759i)9-s + (1.48 + 1.67i)10-s + (−0.709 + 0.890i)11-s + (−1.94 − 0.443i)12-s + (−1.60 − 1.27i)13-s + (−2.64 − 0.0449i)14-s + (−2.59 + 3.62i)15-s + (−0.900 + 0.433i)16-s + (0.405 + 0.0925i)17-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (−0.499 + 1.03i)3-s + (0.111 + 0.487i)4-s + (0.985 + 0.171i)5-s + (−0.733 + 0.352i)6-s + (−0.792 + 0.610i)7-s + (−0.153 + 0.318i)8-s + (−0.201 − 0.253i)9-s + (0.469 + 0.529i)10-s + (−0.214 + 0.268i)11-s + (−0.560 − 0.128i)12-s + (−0.444 − 0.354i)13-s + (−0.707 − 0.0120i)14-s + (−0.669 + 0.935i)15-s + (−0.225 + 0.108i)16-s + (0.0983 + 0.0224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.451783 + 1.57399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.451783 + 1.57399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 5 | \( 1 + (-2.20 - 0.383i)T \) |
| 7 | \( 1 + (2.09 - 1.61i)T \) |
good | 3 | \( 1 + (0.864 - 1.79i)T + (-1.87 - 2.34i)T^{2} \) |
| 11 | \( 1 + (0.709 - 0.890i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.60 + 1.27i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.405 - 0.0925i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + (-0.412 + 0.0942i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.276 + 1.20i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 + (-2.53 - 0.577i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (8.98 + 4.32i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.11 - 8.53i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.165 - 0.132i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-12.9 + 2.95i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-3.09 + 1.49i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.440 - 1.93i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 + (-1.57 - 6.91i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.40 + 3.50i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + (-4.28 + 3.41i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 13.0i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 - 4.55iT - 97T^{2} \) |
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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
−11.28239727215298565959849600386, −10.20836077759741372903752316763, −9.764370922098400764645732363362, −8.893518265670391657060869814859, −7.46920509565568563706161667743, −6.41402570111833176298774254312, −5.49845787002986579004841336598, −5.02874480614517534474558827613, −3.63957086874596078518417009114, −2.47010254622560453008305299294,
0.897114310196708927672266014620, 2.20337694465157211386872321806, 3.55463314896595831696565477472, 5.06319977550523590958431637557, 5.95825069237761873776966827985, 6.73214943820759727578459870355, 7.46988115653917292586211047463, 9.064357436736117130558268504620, 9.887859806159365224826434511605, 10.63381184858205778947183444923