L(s) = 1 | − i·2-s − 4-s + (−1.22 + 2.12i)5-s + (−1.62 − 2.09i)7-s + i·8-s + (2.12 + 1.22i)10-s + (3.67 − 2.12i)11-s + (−0.621 + 0.358i)13-s + (−2.09 + 1.62i)14-s + 16-s + (−1.22 + 2.12i)17-s + (−4.24 + 2.44i)19-s + (1.22 − 2.12i)20-s + (−2.12 − 3.67i)22-s + (5.19 + 3i)23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.547 + 0.948i)5-s + (−0.612 − 0.790i)7-s + 0.353i·8-s + (0.670 + 0.387i)10-s + (1.10 − 0.639i)11-s + (−0.172 + 0.0994i)13-s + (−0.558 + 0.433i)14-s + 0.250·16-s + (−0.297 + 0.514i)17-s + (−0.973 + 0.561i)19-s + (0.273 − 0.474i)20-s + (−0.452 − 0.783i)22-s + (1.08 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.031160838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.031160838\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.62 + 2.09i)T \) |
good | 5 | \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.621 - 0.358i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 - 2.44i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.52 + 0.878i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.08iT - 31T^{2} \) |
| 37 | \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.22 - 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + (-12.5 - 7.24i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 - 4.18iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (4.75 + 2.74i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + (7.64 - 13.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.52 + 2.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.74 + 1.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.11825261947084913784028391982, −9.155222651944567145822580628701, −8.449355886271992172813827854965, −7.21138688199879366576397763161, −6.76433265730334975818035080600, −5.71331026461551730525907935975, −4.18340892436498822396267130306, −3.69104136712967956785736267832, −2.80912391257787855643239534409, −1.22544189118304985333708937457,
0.51718394439874704471367590391, 2.33451403757104796606720910326, 3.85089260871354659326795005598, 4.62006759821355296728877910850, 5.45016219190110289359762969243, 6.53804135304181830241361335707, 7.08337740637392311071200272657, 8.229679727668566399662068884692, 9.026664487168148581325507213066, 9.214157703296154639252955285803