| L(s) = 1 | + (0.0728 + 0.356i)2-s + (0.987 + 0.160i)3-s + (1.71 − 0.731i)4-s + (−1.83 + 0.452i)5-s + (0.0146 + 0.363i)6-s + (−1.86 + 3.93i)7-s + (0.800 + 1.15i)8-s + (0.948 + 0.316i)9-s + (−0.295 − 0.621i)10-s + (5.66 − 1.89i)11-s + (1.81 − 0.446i)12-s + (−3.17 + 1.69i)13-s + (−1.53 − 0.379i)14-s + (−1.88 + 0.152i)15-s + (2.23 − 2.32i)16-s + (−1.59 + 3.36i)17-s + ⋯ |
| L(s) = 1 | + (0.0515 + 0.252i)2-s + (0.569 + 0.0926i)3-s + (0.858 − 0.365i)4-s + (−0.820 + 0.202i)5-s + (0.00598 + 0.148i)6-s + (−0.704 + 1.48i)7-s + (0.282 + 0.409i)8-s + (0.316 + 0.105i)9-s + (−0.0932 − 0.196i)10-s + (1.70 − 0.570i)11-s + (0.523 − 0.129i)12-s + (−0.881 + 0.471i)13-s + (−0.411 − 0.101i)14-s + (−0.486 + 0.0392i)15-s + (0.557 − 0.580i)16-s + (−0.387 + 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57394 + 0.970923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57394 + 0.970923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (3.17 - 1.69i)T \) |
| good | 2 | \( 1 + (-0.0728 - 0.356i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (1.83 - 0.452i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (1.86 - 3.93i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-5.66 + 1.89i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (1.59 - 3.36i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-3.47 - 6.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.85 + 3.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.96 + 9.60i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-2.16 - 1.13i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-4.52 + 2.86i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (3.72 + 0.604i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-4.47 - 2.83i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (0.855 + 7.04i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-3.51 - 5.09i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (3.77 + 3.93i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (10.0 + 0.812i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 2.74i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (3.91 - 4.80i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (4.90 + 4.34i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (1.17 + 9.64i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-1.59 + 4.20i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (0.0240 - 0.0416i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.631 - 2.18i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−11.33040219591934344132832426915, −9.991094415381030853623032796999, −9.271059860891743834099188975099, −8.368329947858211599303034925187, −7.43163727980823812636833754035, −6.37057907821113546189486110821, −5.83933439666405117162479942758, −4.15710177840998352529709950185, −3.09731355099244072881505975435, −1.93393571460671746291937410293,
1.11244310056459379421106130285, 2.92315572340853041751841890465, 3.74272146495826095106739061282, 4.62082719284697607121298604128, 6.73432457501395248320621298619, 7.14264414746025221392517316895, 7.69253141032380063889519611585, 9.160824813349342150372004024258, 9.789528587907351781856332387630, 10.90552025499080525646101615151
