L(s) = 1 | − 2.82·2-s − 5.19i·3-s + 8.00·4-s − 10.0i·5-s + 14.6i·6-s − 22.6·8-s − 27·9-s + 28.4i·10-s + 49.7·11-s − 41.5i·12-s + 116. i·13-s − 52.2·15-s + 64.0·16-s + 273. i·17-s + 76.3·18-s − 12.1i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.402i·5-s + 0.408i·6-s − 0.353·8-s − 0.333·9-s + 0.284i·10-s + 0.410·11-s − 0.288i·12-s + 0.689i·13-s − 0.232·15-s + 0.250·16-s + 0.947i·17-s + 0.235·18-s − 0.0336i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.257186878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257186878\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82T \) |
| 3 | \( 1 + 5.19iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 10.0iT - 625T^{2} \) |
| 11 | \( 1 - 49.7T + 1.46e4T^{2} \) |
| 13 | \( 1 - 116. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 273. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 12.1iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 13.6T + 2.79e5T^{2} \) |
| 29 | \( 1 + 485.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.79e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 533.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.68e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 378.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 522. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.57e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 27.3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.12e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 7.75e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 7.55e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.18e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.59e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.07e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.36e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.46e3iT - 8.85e7T^{2} \) |
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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
−11.10469427424694462172793766417, −10.21030114528764522575846629311, −8.954993019450655051185311354350, −8.530661766902996864623024442146, −7.25682376020747588852103422522, −6.52882249315902988339859029309, −5.28178496288405929187642340010, −3.72258684847004560053953785995, −2.08523268710456227891383562807, −0.991466836901722769671667721687,
0.62215387501445600771856168370, 2.44678918242111501394251710154, 3.63112092067383465311762007460, 5.08923829899768023654150949652, 6.28075204601915299609040429498, 7.35119253184163054146193499316, 8.338414457308145632377465702680, 9.396183621835349395656681549198, 10.02209764333502558882100821346, 11.03940105662616940295363105623