L(s) = 1 | + i·2-s + (1.51 − 0.840i)3-s − 4-s + (0.5 − 0.866i)5-s + (0.840 + 1.51i)6-s + (−1.15 − 2.38i)7-s − i·8-s + (1.58 − 2.54i)9-s + (0.866 + 0.5i)10-s + (−1.81 + 1.04i)11-s + (−1.51 + 0.840i)12-s + (0.413 − 0.238i)13-s + (2.38 − 1.15i)14-s + (0.0292 − 1.73i)15-s + 16-s + (1.44 − 2.49i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.874 − 0.485i)3-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.343 + 0.618i)6-s + (−0.434 − 0.900i)7-s − 0.353i·8-s + (0.528 − 0.848i)9-s + (0.273 + 0.158i)10-s + (−0.546 + 0.315i)11-s + (−0.437 + 0.242i)12-s + (0.114 − 0.0662i)13-s + (0.636 − 0.307i)14-s + (0.00755 − 0.447i)15-s + 0.250·16-s + (0.349 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68720 - 0.595655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68720 - 0.595655i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.51 + 0.840i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.15 + 2.38i)T \) |
good | 11 | \( 1 + (1.81 - 1.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.413 + 0.238i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 2.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.03 + 2.32i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.75 + 1.01i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.74 - 1.58i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.30iT - 31T^{2} \) |
| 37 | \( 1 + (1.93 + 3.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.03 - 6.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.96 - 5.14i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 0.105T + 47T^{2} \) |
| 53 | \( 1 + (-4.27 - 2.46i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 + 4.29iT - 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 + (-5.64 - 3.26i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + (5.58 - 9.68i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.59 - 11.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.98 + 4.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.984800915540305767516286582946, −9.634310496328074215362217855691, −8.597797040626877085631303522031, −7.71114415603738788705062243234, −7.17297938128807294407797590684, −6.21497544650020950168855157233, −4.98390623252152323745485817613, −3.91036839261485520532752412021, −2.72419405909903895977869734245, −0.941302265196472342980017207479,
1.91685690041785555534878035825, 2.97860878961055757483949218349, 3.66389240952283408735980431230, 5.07307845101092921152021970230, 5.95788764343995725322506491726, 7.39180559926070563723070635133, 8.418050418073580177206240423993, 8.995686764904920986566304710440, 10.07237677522386388967933918025, 10.30263239487549908260942813009