L(s) = 1 | + (2.44 + 1.41i)2-s + (3.70 + 2.13i)3-s + (3.99 + 6.92i)4-s + (−17.7 + 30.7i)5-s + (6.04 + 10.4i)6-s + 57.6·7-s + 22.6i·8-s + (−31.3 − 54.3i)9-s + (−87.0 + 50.2i)10-s + 144.·11-s + 34.2i·12-s + (−165. + 95.5i)13-s + (141. + 81.5i)14-s + (−131. + 75.9i)15-s + (−32.0 + 55.4i)16-s + (273. − 474. i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.411 + 0.237i)3-s + (0.249 + 0.433i)4-s + (−0.710 + 1.23i)5-s + (0.168 + 0.291i)6-s + 1.17·7-s + 0.353i·8-s + (−0.387 − 0.670i)9-s + (−0.870 + 0.502i)10-s + 1.19·11-s + 0.237i·12-s + (−0.979 + 0.565i)13-s + (0.720 + 0.415i)14-s + (−0.584 + 0.337i)15-s + (−0.125 + 0.216i)16-s + (0.947 − 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.76546 + 1.25497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76546 + 1.25497i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.44 - 1.41i)T \) |
| 19 | \( 1 + (208. + 294. i)T \) |
good | 3 | \( 1 + (-3.70 - 2.13i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (17.7 - 30.7i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 - 57.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 144.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (165. - 95.5i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-273. + 474. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-97.8 - 169. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (16.6 - 9.62i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + 933. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.97e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.40e3 - 814. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.10e3 - 1.90e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (281. + 487. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (922. - 532. i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.31e3 - 758. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.67e3 + 4.63e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (4.36e3 - 2.52e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (1.23e3 + 711. i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-2.82e3 + 4.88e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-532. - 307. i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.74e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-4.64e3 + 2.68e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (8.50e3 + 4.90e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−15.18310704714278151263865403919, −14.65688036462626301076667219945, −14.04809754445339153712203803657, −11.77153344209382606124287554547, −11.45886855580479690016987867163, −9.422983987626311582979956472008, −7.76653034623996765795538272588, −6.67911489301189218524089048580, −4.54647330676985360873963158767, −3.00415703172333605966638594415,
1.57852801958087679408787028686, 4.08505322674806807497671467438, 5.38072745417983287687553269308, 7.80005247838491675554654747209, 8.687286456596363064817670858559, 10.66198358319655589442251174405, 12.06505593384754033165712932623, 12.66326807102820409842510179489, 14.30634432168646351084710971743, 14.84154759254081542115685782227