| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (−1.87 − 0.684i)5-s + (0.266 + 0.223i)6-s + (0.879 − 1.52i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 2.83i)9-s + (−0.347 + 1.96i)10-s + (−2.11 − 3.66i)11-s + (0.173 − 0.300i)12-s + (−0.815 − 0.684i)13-s + (−1.65 − 0.601i)14-s + (0.652 − 0.237i)15-s + (0.766 − 0.642i)16-s + (1.23 + 7.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.153 + 0.128i)3-s + (−0.469 + 0.171i)4-s + (−0.840 − 0.305i)5-s + (0.108 + 0.0911i)6-s + (0.332 − 0.575i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.945i)9-s + (−0.109 + 0.622i)10-s + (−0.637 − 1.10i)11-s + (0.0501 − 0.0868i)12-s + (−0.226 − 0.189i)13-s + (−0.441 − 0.160i)14-s + (0.168 − 0.0613i)15-s + (0.191 − 0.160i)16-s + (0.300 + 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228572 + 0.233929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228572 + 0.233929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.266 - 0.223i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (1.87 + 0.684i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.879 + 1.52i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.815 + 0.684i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 7.02i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (3.53 - 1.28i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.10 - 6.27i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (4.41 - 7.64i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 + (1.43 - 1.20i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.47 + 1.26i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.638 - 3.61i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (9.29 - 3.38i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.26 - 7.18i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.98 - 1.81i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 11.4i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.65 - 0.965i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.607 + 0.509i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.12 + 4.30i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.754 - 1.30i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.12 + 7.65i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.326 + 1.85i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.83949970699883073635339665311, −10.11642477512335014007544426359, −8.755568612208037222032320787791, −8.070123347802763231608825824408, −7.61288556527227448486531689806, −5.97411345674208741213821719746, −5.00523730353697710939814009630, −4.06148619852762989937823851184, −3.13639733658937900151648796607, −1.54473844178841513247247095117,
0.18148679762108120056524935735, 2.38474138721532130223687871770, 3.81732368892598222459900422812, 4.83827359827261362778886308781, 5.77485778081367932885985263654, 6.84572681095762078884988058561, 7.56107164828938478549774923227, 8.180096411515658938536483276035, 9.497837113705817775246926021484, 9.746018045879543855532512053792
