L(s) = 1 | + (−3 + 1.73i)2-s + (3.5 + 6.06i)3-s + (2 − 3.46i)4-s − 13.8i·5-s + (−21 − 12.1i)6-s + (19.5 + 11.2i)7-s − 13.8i·8-s + (−11 + 19.0i)9-s + (23.9 + 41.5i)10-s + (−19.5 + 11.2i)11-s + 28.0·12-s + (−13 − 45.0i)13-s − 78·14-s + (84 − 48.4i)15-s + (39.9 + 69.2i)16-s + (−13.5 + 23.3i)17-s + ⋯ |
L(s) = 1 | + (−1.06 + 0.612i)2-s + (0.673 + 1.16i)3-s + (0.250 − 0.433i)4-s − 1.23i·5-s + (−1.42 − 0.824i)6-s + (1.05 + 0.607i)7-s − 0.612i·8-s + (−0.407 + 0.705i)9-s + (0.758 + 1.31i)10-s + (−0.534 + 0.308i)11-s + 0.673·12-s + (−0.277 − 0.960i)13-s − 1.48·14-s + (1.44 − 0.834i)15-s + (0.624 + 1.08i)16-s + (−0.192 + 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.566359 + 0.437474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.566359 + 0.437474i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (13 + 45.0i)T \) |
good | 2 | \( 1 + (3 - 1.73i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 13.8iT - 125T^{2} \) |
| 7 | \( 1 + (-19.5 - 11.2i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (19.5 - 11.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (13.5 - 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (76.5 + 44.1i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (28.5 + 49.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-34.5 - 59.7i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 72.7iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (34.5 - 19.9i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (340.5 - 196. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-42.5 + 73.6i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 342. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 426T + 1.48e5T^{2} \) |
| 59 | \( 1 + (16.5 + 9.52i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-8.5 + 14.7i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-142.5 + 82.2i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-505.5 - 291. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 1.00e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 426. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-265.5 + 153. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.06e3 - 617. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−19.88636005144530613870746315734, −18.07827698741142481588090997022, −16.99354970851497277558510659429, −15.75822802309119482862262646720, −14.96920099474815101760947911563, −12.71056452343602839582459843870, −10.30179298841582890889348698943, −8.868693986454882017884086778155, −8.231085512020065232240113341638, −4.80198786544075143735648168099,
2.11014574367122488708249141919, 7.18864249379947825917987191008, 8.360288387817286688492718603371, 10.39776468923365132375400991003, 11.57781508819803023179857827817, 13.78044923184104415161506893934, 14.59910339934377349528497014381, 17.26214822321643194606012520434, 18.44175080340776231630660574013, 18.85927168084700490843340412488