| L(s) = 1 | + (2.73 − 2.92i)2-s + (−1.08 − 15.9i)4-s + (−19.5 + 33.8i)5-s + (−10.5 + 6.10i)7-s + (−49.6 − 40.4i)8-s + (45.5 + 149. i)10-s + (−96.1 + 55.5i)11-s + (−104. + 180. i)13-s + (−11.0 + 47.5i)14-s + (−253. + 34.7i)16-s − 93.3·17-s − 26.8i·19-s + (561. + 275. i)20-s + (−100. + 432. i)22-s + (757. + 437. i)23-s + ⋯ |
| L(s) = 1 | + (0.682 − 0.730i)2-s + (−0.0680 − 0.997i)4-s + (−0.781 + 1.35i)5-s + (−0.215 + 0.124i)7-s + (−0.775 − 0.631i)8-s + (0.455 + 1.49i)10-s + (−0.794 + 0.458i)11-s + (−0.618 + 1.07i)13-s + (−0.0562 + 0.242i)14-s + (−0.990 + 0.135i)16-s − 0.323·17-s − 0.0744i·19-s + (1.40 + 0.687i)20-s + (−0.207 + 0.893i)22-s + (1.43 + 0.826i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.533653 + 0.638772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.533653 + 0.638772i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 + 2.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (19.5 - 33.8i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (10.5 - 6.10i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (96.1 - 55.5i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (104. - 180. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 93.3T + 8.35e4T^{2} \) |
| 19 | \( 1 + 26.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-757. - 437. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (650. + 1.12e3i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-593. - 342. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-39.0 + 67.6i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.40e3 - 811. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.99e3 + 1.15e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 1.31e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (4.81e3 + 2.78e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.09e3 + 1.88e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (213. + 123. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 4.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-4.48e3 + 2.59e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.62e3 + 936. i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 1.16e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-2.86e3 - 4.97e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.26755778698928787762341898063, −12.04669421110214470744974903119, −11.29262392606600431020735383331, −10.43241162128717605563044546562, −9.343698595606786218444110489455, −7.44757221286683241915498021598, −6.52506931168487381071292922793, −4.86246837723885878417865651971, −3.49345262076048686694947453158, −2.33304381480465380532885384453,
0.28169483272763404593417492724, 3.19023194676080291335178733449, 4.69201490262310496336733379630, 5.45360272959469313184432294391, 7.16829874516147494968229214838, 8.190049701227254377350929102333, 8.963805136133654056058834250669, 10.76429783478544126079978793342, 12.16632885225982800723460645091, 12.77930752408363651411988118395
