| L(s) = 1 | + (2.77 + 15.7i)2-s + (15.1 − 12.7i)3-s + (−240. + 87.5i)4-s + (−8.62 − 3.13i)5-s + (242. + 203. i)6-s + (80.6 − 139. i)7-s + (−2.04e3 − 3.54e3i)8-s + (−3.35e3 + 1.89e4i)9-s + (25.5 − 144. i)10-s + (−2.92e4 − 5.05e4i)11-s + (−2.52e3 + 4.38e3i)12-s + (−2.44e4 − 2.05e4i)13-s + (2.42e3 + 882. i)14-s + (−170. + 62.0i)15-s + (5.02e4 − 4.21e4i)16-s + (−8.27e4 − 4.69e5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.107 − 0.0905i)3-s + (−0.469 + 0.171i)4-s + (−0.00617 − 0.00224i)5-s + (0.0763 + 0.0640i)6-s + (0.0126 − 0.0219i)7-s + (−0.176 − 0.306i)8-s + (−0.170 + 0.965i)9-s + (0.000806 − 0.00457i)10-s + (−0.601 − 1.04i)11-s + (−0.0352 + 0.0610i)12-s + (−0.237 − 0.199i)13-s + (0.0168 + 0.00614i)14-s + (−0.000869 + 0.000316i)15-s + (0.191 − 0.160i)16-s + (−0.240 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000992 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.000992 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.497717 - 0.498211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497717 - 0.498211i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.77 - 15.7i)T \) |
| 19 | \( 1 + (4.96e5 - 2.75e5i)T \) |
| good | 3 | \( 1 + (-15.1 + 12.7i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (8.62 + 3.13i)T + (1.49e6 + 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-80.6 + 139. i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.92e4 + 5.05e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (2.44e4 + 2.05e4i)T + (1.84e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (8.27e4 + 4.69e5i)T + (-1.11e11 + 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-1.26e6 + 4.61e5i)T + (1.37e12 - 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-1.14e6 + 6.48e6i)T + (-1.36e13 - 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-2.08e6 + 3.61e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 4.09e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + (8.99e6 - 7.54e6i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + (3.68e7 + 1.34e7i)T + (3.85e14 + 3.23e14i)T^{2} \) |
| 47 | \( 1 + (-1.34e6 + 7.60e6i)T + (-1.05e15 - 3.82e14i)T^{2} \) |
| 53 | \( 1 + (7.90e7 - 2.87e7i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (-2.86e7 - 1.62e8i)T + (-8.14e15 + 2.96e15i)T^{2} \) |
| 61 | \( 1 + (7.99e6 - 2.90e6i)T + (8.95e15 - 7.51e15i)T^{2} \) |
| 67 | \( 1 + (2.69e7 - 1.52e8i)T + (-2.55e16 - 9.30e15i)T^{2} \) |
| 71 | \( 1 + (2.34e8 + 8.54e7i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-1.98e7 + 1.66e7i)T + (1.02e16 - 5.79e16i)T^{2} \) |
| 79 | \( 1 + (-4.94e7 + 4.14e7i)T + (2.08e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-1.99e8 + 3.45e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.69e8 + 1.41e8i)T + (6.08e16 + 3.45e17i)T^{2} \) |
| 97 | \( 1 + (2.12e8 + 1.20e9i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−13.87972307116687723828274377201, −13.27004663961017294590909439962, −11.59259528664710749382218184980, −10.21379720354115376445657033724, −8.574022634590286368560665386391, −7.61096131031398133590532228506, −6.00728214404554925628335010376, −4.67604502598846800945073007393, −2.67083065580029770770749178126, −0.23449062007130763034765410854,
1.72395305831242158717092659813, 3.40024973120543776148476002689, 4.90508141956243857650137106264, 6.72885796168191134452992268909, 8.558195300136424546991478721450, 9.763867554921490662722766323636, 10.92760543344680330733309062715, 12.30252855873917306066108464270, 13.08510453501941942805438463871, 14.68386591095563059999227854785
