| L(s) = 1 | + (−2.48 + 0.200i)2-s + (−2.64 + 1.67i)3-s + (4.16 − 0.677i)4-s + (−0.748 + 0.663i)5-s + (6.23 − 4.68i)6-s + (−0.323 − 0.336i)7-s + (−5.38 + 1.32i)8-s + (2.90 − 6.11i)9-s + (1.72 − 1.79i)10-s + (2.76 + 5.83i)11-s + (−9.88 + 8.75i)12-s + (2.69 + 2.39i)13-s + (0.871 + 0.772i)14-s + (0.869 − 3.00i)15-s + (5.11 − 1.70i)16-s + (−3.18 − 3.31i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 + 0.141i)2-s + (−1.52 + 0.964i)3-s + (2.08 − 0.338i)4-s + (−0.334 + 0.296i)5-s + (2.54 − 1.91i)6-s + (−0.122 − 0.127i)7-s + (−1.90 + 0.469i)8-s + (0.967 − 2.03i)9-s + (0.546 − 0.568i)10-s + (0.834 + 1.75i)11-s + (−2.85 + 2.52i)12-s + (0.747 + 0.663i)13-s + (0.232 + 0.206i)14-s + (0.224 − 0.775i)15-s + (1.27 − 0.427i)16-s + (−0.773 − 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.201503 - 0.0278740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.201503 - 0.0278740i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.748 - 0.663i)T \) |
| 13 | \( 1 + (-2.69 - 2.39i)T \) |
| good | 2 | \( 1 + (2.48 - 0.200i)T + (1.97 - 0.320i)T^{2} \) |
| 3 | \( 1 + (2.64 - 1.67i)T + (1.28 - 2.71i)T^{2} \) |
| 7 | \( 1 + (0.323 + 0.336i)T + (-0.281 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-2.76 - 5.83i)T + (-6.95 + 8.52i)T^{2} \) |
| 17 | \( 1 + (3.18 + 3.31i)T + (-0.684 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.18 + 3.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.20 + 2.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.98 - 0.563i)T + (28.6 - 4.65i)T^{2} \) |
| 31 | \( 1 + (0.685 + 5.64i)T + (-30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (6.51 + 2.77i)T + (25.6 + 26.6i)T^{2} \) |
| 41 | \( 1 + (1.52 - 0.962i)T + (17.5 - 37.0i)T^{2} \) |
| 43 | \( 1 + (-4.82 + 2.05i)T + (29.7 - 31.0i)T^{2} \) |
| 47 | \( 1 + (-0.781 - 2.06i)T + (-35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (11.1 - 2.74i)T + (46.9 - 24.6i)T^{2} \) |
| 59 | \( 1 + (-0.674 - 0.225i)T + (47.1 + 35.4i)T^{2} \) |
| 61 | \( 1 + (-0.852 - 2.94i)T + (-51.5 + 32.6i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 1.70i)T + (63.5 + 21.2i)T^{2} \) |
| 71 | \( 1 + (-0.127 - 3.15i)T + (-70.7 + 5.71i)T^{2} \) |
| 73 | \( 1 + (-3.44 + 4.98i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (4.52 + 11.9i)T + (-59.1 + 52.3i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 5.30i)T + (47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + (-3.97 + 6.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.83 + 18.7i)T + (-89.2 - 38.0i)T^{2} \) |
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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
−10.13152493966331014915677586768, −9.288274653854320219428828784697, −9.043883069078876678235099931604, −7.37751936333070718794541985752, −6.83513174249027071582299264674, −6.24513601377222256749424674886, −4.80305998286537181886995573270, −3.99312803754542371891892001073, −1.92981588214545086237356337390, −0.30533061490442207158614818374,
0.883629283421890653719056245855, 1.70198421350162779609086072958, 3.59173281097282181357265808430, 5.48497081533353603955192751616, 6.25312764314211231621856520331, 6.76776020997406210361644748672, 7.942778502401817482263289312935, 8.387362162864740339296840764967, 9.259740974145493240368624398334, 10.59185370538450908240628443120
