| L(s) = 1 | + (0.437 + 1.34i)2-s + (−0.426 + 0.384i)3-s + (−1.61 + 1.17i)4-s + (2.75 + 4.77i)5-s + (−0.703 − 0.405i)6-s + (0.186 + 1.77i)7-s + (−2.28 − 1.66i)8-s + (−0.906 + 8.62i)9-s + (−5.21 + 5.79i)10-s + (7.06 − 15.8i)11-s + (0.238 − 1.12i)12-s + (−1.57 − 7.42i)13-s + (−2.30 + 1.02i)14-s + (−3.00 − 0.977i)15-s + (1.23 − 3.80i)16-s + (−4.04 − 9.08i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.142 + 0.128i)3-s + (−0.404 + 0.293i)4-s + (0.551 + 0.954i)5-s + (−0.117 − 0.0676i)6-s + (0.0266 + 0.253i)7-s + (−0.286 − 0.207i)8-s + (−0.100 + 0.958i)9-s + (−0.521 + 0.579i)10-s + (0.642 − 1.44i)11-s + (0.0198 − 0.0936i)12-s + (−0.121 − 0.571i)13-s + (−0.164 + 0.0732i)14-s + (−0.200 − 0.0651i)15-s + (0.0772 − 0.237i)16-s + (−0.237 − 0.534i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0504 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0504 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.947189 + 0.900594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.947189 + 0.900594i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.437 - 1.34i)T \) |
| 31 | \( 1 + (30.5 - 5.51i)T \) |
| good | 3 | \( 1 + (0.426 - 0.384i)T + (0.940 - 8.95i)T^{2} \) |
| 5 | \( 1 + (-2.75 - 4.77i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.186 - 1.77i)T + (-47.9 + 10.1i)T^{2} \) |
| 11 | \( 1 + (-7.06 + 15.8i)T + (-80.9 - 89.9i)T^{2} \) |
| 13 | \( 1 + (1.57 + 7.42i)T + (-154. + 68.7i)T^{2} \) |
| 17 | \( 1 + (4.04 + 9.08i)T + (-193. + 214. i)T^{2} \) |
| 19 | \( 1 + (-21.8 - 4.64i)T + (329. + 146. i)T^{2} \) |
| 23 | \( 1 + (-14.1 + 19.5i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (31.0 - 10.0i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 + (-29.5 - 17.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (27.5 - 30.5i)T + (-175. - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (4.74 - 22.3i)T + (-1.68e3 - 752. i)T^{2} \) |
| 47 | \( 1 + (-6.56 + 20.2i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (64.5 + 6.78i)T + (2.74e3 + 584. i)T^{2} \) |
| 59 | \( 1 + (14.9 + 16.6i)T + (-363. + 3.46e3i)T^{2} \) |
| 61 | \( 1 - 78.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (12.3 + 21.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 107. i)T + (-4.93e3 - 1.04e3i)T^{2} \) |
| 73 | \( 1 + (-31.6 + 71.0i)T + (-3.56e3 - 3.96e3i)T^{2} \) |
| 79 | \( 1 + (0.245 + 0.551i)T + (-4.17e3 + 4.63e3i)T^{2} \) |
| 83 | \( 1 + (95.5 + 85.9i)T + (720. + 6.85e3i)T^{2} \) |
| 89 | \( 1 + (-47.7 - 65.7i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-86.3 + 62.7i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−14.86388086437213762472435439523, −14.05809308339684634575353453763, −13.23873500852708322389653561771, −11.51155676315506861184546945732, −10.54256311893174019722155608245, −9.098947128219292306107913861450, −7.71205560635646152796700990974, −6.34454984548004938999152322194, −5.26240120140118032414697061043, −3.08666616952047043372187584978,
1.53300214422095946538772737313, 4.02613356043181603303559441923, 5.45447775367016057774293591374, 7.11442104138012508415875030521, 9.218718588161682913861033852788, 9.579150186930988612613177114870, 11.34944338534929439215068967184, 12.35247186407802039488171121138, 13.08659667867950035608954502672, 14.32502539087082724908675822359
