L(s) = 1 | + (0.895 + 1.09i)2-s + (−0.395 + 1.96i)4-s + (1.5 + 0.866i)5-s + (−2.29 + 1.32i)7-s + (−2.49 + 1.32i)8-s + (0.395 + 2.41i)10-s + (2.29 − 1.32i)11-s + 3.46i·13-s + (−3.49 − 1.32i)14-s + (−3.68 − 1.55i)16-s + (6 − 3.46i)17-s + (−2.29 + 2.59i)20-s + (3.49 + 1.32i)22-s + (−4.58 − 2.64i)23-s + (−1 − 1.73i)25-s + (−3.79 + 3.10i)26-s + ⋯ |
L(s) = 1 | + (0.633 + 0.773i)2-s + (−0.197 + 0.980i)4-s + (0.670 + 0.387i)5-s + (−0.866 + 0.499i)7-s + (−0.883 + 0.467i)8-s + (0.125 + 0.764i)10-s + (0.690 − 0.398i)11-s + 0.960i·13-s + (−0.935 − 0.353i)14-s + (−0.921 − 0.387i)16-s + (1.45 − 0.840i)17-s + (−0.512 + 0.580i)20-s + (0.746 + 0.282i)22-s + (−0.955 − 0.551i)23-s + (−0.200 − 0.346i)25-s + (−0.743 + 0.608i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08336 + 1.32192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08336 + 1.32192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.895 - 1.09i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.29 - 1.32i)T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.32i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-6 + 3.46i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.58 + 2.64i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (-2.29 - 3.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + (-4.58 + 7.93i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.5 + 6.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.87 - 11.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9 + 5.19i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.87 - 3.96i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + (9 + 5.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−12.13956560328256285383956800252, −11.95659398349140299614136947026, −10.20342115044766301392937901975, −9.332823616632220215454028671318, −8.377610455930587102247578440347, −6.94923394715054589566559481010, −6.31150779962359825559277366615, −5.38871756398969492864518484788, −3.87518289545552380109473090908, −2.65247751486771591716730507980,
1.32667195087044625802145231329, 3.08933033022159111907753648818, 4.19176773908156475115189996166, 5.62591232774927155472967968270, 6.29992983151599447495063860013, 7.85539361066914366901467917181, 9.442974547190303328699987058312, 9.891964026150207663478009508633, 10.73185401262585542193012581796, 12.09543811588003594651656085685