L(s) = 1 | + (−0.998 − 0.725i)2-s + (−0.112 + 0.346i)3-s + (−0.147 − 0.453i)4-s + (0.363 − 0.264i)6-s + (0.798 + 2.45i)7-s + (−0.944 + 2.90i)8-s + (2.31 + 1.68i)9-s + (3.12 − 1.12i)11-s + 0.173·12-s + (−2.23 − 1.62i)13-s + (0.985 − 3.03i)14-s + (2.27 − 1.65i)16-s + (3.11 − 2.26i)17-s + (−1.09 − 3.36i)18-s + (−0.0857 + 0.264i)19-s + ⋯ |
L(s) = 1 | + (−0.705 − 0.512i)2-s + (−0.0649 + 0.199i)3-s + (−0.0737 − 0.226i)4-s + (0.148 − 0.107i)6-s + (0.301 + 0.929i)7-s + (−0.333 + 1.02i)8-s + (0.773 + 0.561i)9-s + (0.940 − 0.339i)11-s + 0.0501·12-s + (−0.619 − 0.449i)13-s + (0.263 − 0.810i)14-s + (0.569 − 0.414i)16-s + (0.755 − 0.549i)17-s + (−0.257 − 0.793i)18-s + (−0.0196 + 0.0605i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918855 - 0.119502i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918855 - 0.119502i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (-3.12 + 1.12i)T \) |
good | 2 | \( 1 + (0.998 + 0.725i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.112 - 0.346i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.798 - 2.45i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2.23 + 1.62i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.11 + 2.26i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0857 - 0.264i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 8.40T + 23T^{2} \) |
| 29 | \( 1 + (-1.02 - 3.16i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.456 + 0.331i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.161 - 0.497i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.57 - 4.86i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.54T + 43T^{2} \) |
| 47 | \( 1 + (-1.52 + 4.68i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.05 + 5.12i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.31 - 7.13i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (11.4 - 8.33i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + (-6.79 + 4.93i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.02 - 12.3i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (7.85 + 5.70i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.66 + 1.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + (8.81 + 6.40i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.64005046431016802543818849295, −10.83217529251844548093714602626, −9.895025568765427855736762972710, −9.180503144311268666602701435749, −8.325243829669572003217680861874, −7.08507414014932802381079575247, −5.59079542694564168871217820157, −4.81330079976348110882326854446, −2.90032001293306607477962561396, −1.40624364124403549625136251709,
1.17209710502650140339888854046, 3.59605115222321990277565033409, 4.56925358905051659464691104678, 6.48226513447634780560685552814, 7.13227396898255541680298454784, 7.86314535330821673275947299939, 9.151764634492492471853569179265, 9.718446652736048189397722497496, 10.82245246954732165964716658434, 12.11299121977565196815811286243