L(s) = 1 | + (−4.85 + 2.80i)2-s + (−2.59 − 1.5i)3-s + (11.7 − 20.3i)4-s + (2.40 − 10.9i)5-s + 16.8·6-s + (−9.44 − 15.9i)7-s + 86.8i·8-s + (4.5 + 7.79i)9-s + (18.9 + 59.7i)10-s + (−9.08 + 15.7i)11-s + (−61.0 + 35.2i)12-s − 34.8i·13-s + (90.5 + 50.9i)14-s + (−22.6 + 24.7i)15-s + (−149. − 259. i)16-s + (−5.05 − 2.91i)17-s + ⋯ |
L(s) = 1 | + (−1.71 + 0.991i)2-s + (−0.499 − 0.288i)3-s + (1.46 − 2.54i)4-s + (0.214 − 0.976i)5-s + 1.14·6-s + (−0.509 − 0.860i)7-s + 3.83i·8-s + (0.166 + 0.288i)9-s + (0.599 + 1.89i)10-s + (−0.248 + 0.431i)11-s + (−1.46 + 0.847i)12-s − 0.744i·13-s + (1.72 + 0.972i)14-s + (−0.389 + 0.426i)15-s + (−2.34 − 4.05i)16-s + (−0.0721 − 0.0416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0145253 - 0.113274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0145253 - 0.113274i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.59 + 1.5i)T \) |
| 5 | \( 1 + (-2.40 + 10.9i)T \) |
| 7 | \( 1 + (9.44 + 15.9i)T \) |
good | 2 | \( 1 + (4.85 - 2.80i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (9.08 - 15.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 34.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (5.05 + 2.91i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-30.5 - 52.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (56.5 - 32.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (36.9 - 63.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-52.5 + 30.3i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (24.8 - 14.3i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (366. + 211. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-220. + 382. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-48.3 - 83.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (337. + 194. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 343.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-607. - 350. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-40.8 - 70.7i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.35e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (357. + 619. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 332. iT - 9.12e5T^{2} \) |
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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
−12.77918275815483672786243005119, −11.32768761058707187084851014044, −10.15647087759273056801496207693, −9.587724934938531564001641054450, −8.210262934742154807542017522318, −7.43542134808816678101065968742, −6.23678076434273599737418293630, −5.17341384305183254698836028827, −1.46215409662059324112173208883, −0.11826079334754845452622018706,
2.20100007555087372818914830654, 3.46224526889618237962526792289, 6.25790951364658634150779476772, 7.31675433777678137292078918794, 8.775552893090081484857822728895, 9.622919130361512788356954077116, 10.49836150687809494759151886349, 11.39679484980722771829678149904, 12.04382980995760484279065509798, 13.33577725341586341721516555452