| L(s) = 1 | + (−1.61 − 0.824i)2-s + (3.42 + 0.541i)3-s + (−0.411 − 0.567i)4-s + (4.00 − 2.99i)5-s + (−5.08 − 3.69i)6-s + (−8.06 + 8.06i)7-s + (1.33 + 8.43i)8-s + (2.84 + 0.924i)9-s + (−8.94 + 1.55i)10-s + (−1.43 − 4.43i)11-s + (−1.10 − 2.16i)12-s + (−7.37 + 3.75i)13-s + (19.6 − 6.40i)14-s + (15.3 − 8.09i)15-s + (3.92 − 12.0i)16-s + (16.4 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.809 − 0.412i)2-s + (1.14 + 0.180i)3-s + (−0.102 − 0.141i)4-s + (0.800 − 0.599i)5-s + (−0.848 − 0.616i)6-s + (−1.15 + 1.15i)7-s + (0.166 + 1.05i)8-s + (0.316 + 0.102i)9-s + (−0.894 + 0.155i)10-s + (−0.130 − 0.402i)11-s + (−0.0918 − 0.180i)12-s + (−0.567 + 0.289i)13-s + (1.40 − 0.457i)14-s + (1.02 − 0.539i)15-s + (0.245 − 0.755i)16-s + (0.969 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.792310 - 0.217368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792310 - 0.217368i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.00 + 2.99i)T \) |
| good | 2 | \( 1 + (1.61 + 0.824i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-3.42 - 0.541i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (8.06 - 8.06i)T - 49iT^{2} \) |
| 11 | \( 1 + (1.43 + 4.43i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (7.37 - 3.75i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-16.4 + 2.60i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (2.60 - 3.58i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-1.57 + 3.09i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (14.6 + 20.1i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-39.2 - 28.5i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (12.2 + 23.9i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-10.7 + 33.1i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (13.5 + 13.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.94 - 56.4i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (15.9 + 2.51i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-50.4 - 16.3i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-25.1 - 77.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-3.74 + 0.593i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (23.0 - 16.7i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (19.4 - 38.1i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-46.6 - 64.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (9.69 + 61.2i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-91.0 + 29.5i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (22.8 - 143. i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−17.49311957262872273209229326111, −16.14836795914171243805525822764, −14.67688116524244547093657282395, −13.63873119851696489319735280285, −12.20516414258424704035324887137, −9.990155897891212833059232893121, −9.286706498448610616554362517031, −8.442083970512024467102230264263, −5.68204057349567982472666962103, −2.57830703713719552368026270411,
3.29210839300940802357526598564, 6.80320545103478328259686716882, 7.87364214351724092424156833705, 9.525895034682326337327757608254, 10.14523961610872954741333077986, 12.95325226725529601508001966996, 13.71082521248575502763307964358, 14.97689412742107120524043449758, 16.59688372500257068806328229742, 17.41440179037054602484524314335
