L(s) = 1 | + (−0.0871 + 0.996i)2-s + (−1.06 − 1.36i)3-s + (−0.984 − 0.173i)4-s + (1.95 + 1.08i)5-s + (1.45 − 0.940i)6-s + (−1.28 + 0.343i)7-s + (0.258 − 0.965i)8-s + (−0.737 + 2.90i)9-s + (−1.25 + 1.85i)10-s + (−4.64 + 2.68i)11-s + (0.810 + 1.53i)12-s + (1.47 − 3.17i)13-s + (−0.230 − 1.30i)14-s + (−0.586 − 3.82i)15-s + (0.939 + 0.342i)16-s + (−0.458 + 5.23i)17-s + ⋯ |
L(s) = 1 | + (−0.0616 + 0.704i)2-s + (−0.614 − 0.789i)3-s + (−0.492 − 0.0868i)4-s + (0.873 + 0.487i)5-s + (0.593 − 0.383i)6-s + (−0.485 + 0.129i)7-s + (0.0915 − 0.341i)8-s + (−0.245 + 0.969i)9-s + (−0.397 + 0.585i)10-s + (−1.40 + 0.808i)11-s + (0.233 + 0.441i)12-s + (0.409 − 0.879i)13-s + (−0.0616 − 0.349i)14-s + (−0.151 − 0.988i)15-s + (0.234 + 0.0855i)16-s + (−0.111 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.324391 + 0.687792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.324391 + 0.687792i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0871 - 0.996i)T \) |
| 3 | \( 1 + (1.06 + 1.36i)T \) |
| 5 | \( 1 + (-1.95 - 1.08i)T \) |
| 19 | \( 1 + (-4.21 + 1.12i)T \) |
good | 7 | \( 1 + (1.28 - 0.343i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (4.64 - 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.47 + 3.17i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (0.458 - 5.23i)T + (-16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (6.24 - 4.37i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-1.96 - 1.64i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.70 - 8.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.77 + 2.77i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.57 - 4.31i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.480 - 0.336i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-3.09 + 0.270i)T + (46.2 - 8.16i)T^{2} \) |
| 53 | \( 1 + (-1.61 + 1.12i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (11.6 - 9.74i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.33 - 7.59i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.05 + 12.0i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-5.16 + 0.910i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.3 + 5.30i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (-0.195 + 0.538i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.9 + 3.47i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-7.24 + 2.63i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.27 - 0.286i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.60208582422165560558831956116, −10.44289381035030374541765566455, −9.304915936724215241432905698095, −8.023797129633726917981746698589, −7.43672160434141110292106126546, −6.44606444808865299501623127299, −5.71535284280088442321661604433, −5.09029792596537169747860328860, −3.15814021405825584401053272489, −1.74567443075808619193196208686,
0.46324460497519967589216818568, 2.40491866687345759496016091393, 3.63581334645433136177677124704, 4.83446340515815782383740846362, 5.57111819176150151348538788748, 6.46217108633073767074263902099, 8.030486436561068051940356540824, 9.119861920170293763672827504346, 9.696389178743851993994474808138, 10.36046907836866659895316956713